In an answer from a previous question, the Padishah Emperor conspired against the Atreides because he saw Leto as a threat to his reign- but were the Emperor's suspicions justified?
As far as I know, there isn't any evidence either way. Part of the problem is Leto suspects the emperor of conspiring with the Harkonnens from the very beginning, so we don't know what he thought before he was aware of this danger and there just isn't much on Leto in the book period.
This is considering just the original novel. I don't know if the prequel novels go into more detail, but I would be reluctant to consider them even if they do. In the original novel all we have is Leto's popularity in the Landsraad and that he was beginning to train soldiers that could individually match the Sardaukar but no evidence either way that he planned to use either or both to overthrow Shaddam.
To Duke Leto, gaining the Imperial Throne for House Atreides was a possibility.
For Shaddam IV, losing the Throne for House Corrino was an inevitability. The threat to his reign came from Primogeniture, The Emperor knew his dynasty was ending because he had no male heir. Duke Leto wasn't just another suitor, who could ascend through marriage, he was one whom the emperor disliked.
information from the "Prelude to dune" prequels:
Shaddam IV would be the last Corrino Emperor due to the Bene Gesserit taking control of the imperial seed through marriage and contraceptives. The Corrino heiress, Irulan Corrino, was of marriable age to either Leto or Paul, among a great number of other potential suitors. So why not arrange a marriage to Duke Leto the just? Because the Emperor can hold a grudge. Duke Leto and Shaddam IV have a complicated history involving: public opinion following Military campaigns, supporting different sides in a proxy war on IX, and the Duke used blackmail to receive an imperial pardon during a trial. The sordid past between Shaddam IV and Leto is what lead the Emperor to conspire with the Harkonnens.