Did Napoleon plan to invade India?

The answer is a very solid yes although I'd prefer the word Intervene rather than invade.

India was one of the key possessions of British and Indian trade was crucial to British economy at that time. As long as British economy was strong, Britain would have been able to field expeditionary forces to thwart French ambitions worldwide.

So that's where comes the Bold plan to hit the British where it would really hurt: India.

As early as February 1798, Napoleon had presented his report to the directory regarding the proposed invasion of England in which he ruled out an invasion until French Navy could take on the Royal Navy and in the meanwhile he provided the Directory three alternatives, one of which was to launch an Eastern invasion to damage British Indian trade. His words were:

Whatever efforts we make, we shall not for some years gain naval supremacy. To invade England without that supremacy is the most daring and difficult task ever undertaken . . . If, having regard to the present organization of our navy, it seems impossible to gain the necessary promptness of execution, then we must really give up the expedition against England – be satisfied with keeping up the pretence of it – and concentrate all our attention and resources on the Rhine, in order to try to deprive England of Hanover . . . or else undertake an eastern expedition which would menace her trade with the Indies. And if none of these three operations is practicable, I see nothing else for it but to conclude peace.
_{Napoleon: A Life - Andrew Roberts}

Although, at that point, the invasion of Egypt was to severe British mediterranean links with India and in that way it was to hurt their trade with India. Although Napoleon did not rule out using Egypt as a stepping stone to India when talking to his secretary according to Andrew Roberts.

He did demand maps of India from the directory but his "dream" was broken when the Directory directed him that his command was to invade Egypt, nothing more.

His ultimate ambition – or fantasy – may be gauged by his demand for English maps of Bengal and the Ganges from the war ministry, and his request to be accompanied by Citizen Piveron, the former envoy to Britain’s greatest enemy in India, Tipu Sahib, ‘the Tiger of Mysore’. Yet the Directory deflated these dreams; Napoleon was authorized only to invade Egypt and was told to raise the funds himself. He was expected to be back in France in six months.
_{Napoleon: A Life - Andrew Roberts}

Nevertheless it was never really off the table for him as we see the next year, he was still talking to Directory about taking India after his conquest of Egypt.

Once Desaix had routed Murad Bey at the battle of Samhoud in January 1799, captured his flotilla on the Nile and ended the threat from Upper Egypt, Napoleon’s rule extended over almost the whole country. He could now unleash his attack on Jezzar. He told the Directory on the day he left Cairo that he hoped to deny the Royal Navy the use of Levantine ports such as Acre, Haifa and Jaffa, raise the Lebanese and Syrian Christians in revolt against the Turks, and decide later whether to march on Constantinople or India.
_{Napoleon: A Life - Andrew Roberts}

One of the Key goals of Napoleon's invasion of Egypt was to link up with his Indian allies and help them in liberating their homeland against the British. This would not only damage the British economy but also win friendship and trade concessions for the French in Indian subcontinent.

According to Napoleon and Persia by Irdaj Amini:

He [Napoleon] assured the Directoire that "as soon as he had conquered Egypt, he will establish relations with the Indian princes and, together with them, attack the English in their possessions."

According to Tricolor and Crescent by William E. Watson:

Bonaparte wished to establish a French presence in the Middle East, with the ultimate dream of linking with France's ally Tipu Sultan, ruler of Mysore in India.

French Diplomat Talleyrand wrote this report on 13 February 1798:

Having occupied and fortified Egypt, we shall send a force of 15,000 men from Suez to India, to join the forces of Tipu-Sahib and drive away the English

In addition, In 1801, Napoleon and Paul I of Russia formed a mutual plan to invade India and fight the British there. Russian contingent was ordered to advance to India but as soon as they reached the Aral sea, they received the news that Paul I was assassinated and the plan scuttled.

So friendly were Napoleon’s relations with Paul by early 1801 that plans were even drawn up for Masséna to enter Astrakhan with 35,000 men, join up with 35,000 Russians and 50,000 Cossacks, and then cross the Caspian Sea to take Kandahar, from where they would invade India. It was another of Napoleon’s far-fetched Oriental schemes, though not so fantastical as a march from Aleppo.
_{Napoleon: A Life - Andrew Roberts}

Napoleon also formed a Franco-Persian alliance in 1807 during the Napoleonic Wars, with the continuous aim of having an eventual open way to attack British India.

Napoleon's interest in the Middle East and India waned when he finally vanquished Russia at the Battle of Friedland in July 1807, leading to the Treaty of Tilsit, in December 1807. France as an ally of Russia became a strong repellent to both the Ottoman Empire and Persia. Persia then denounced its alliance with France and moved towards Great Britain.These last events rendered the Franco-Indian alliance impossible to pursue further.

In any case, in conclusion, Napoleon desired to help Indians because:

1. He wanted to destroy British trading in Rich Indian subcontinent.
2. He wanted to open another front against the British to cause them to stretch their forces. Distracted British were easier to deal with than composed and focused ones.

Nader Shah only sacked Delhi and then returned to his homeland. He did not intend to rule India. It is unclear whatever motives Napoleon harboured in his heart, on the surface it appears that his sole ambition was to disrupt the British trade and force them to squander their strength in fighting Indians.