Doubtful. Even with his power, Palpatine couldn't anticipate every single thing that occurred. He had only just discovered Anakin and while he had plans to make him his knew apprentice, that was along time ahead. If he really wanted Darth Maul out of the picture then he would have done it himself.
Palpatine despised Darth Maul for not being "quite good enough" and lost interest in him once his gaze turned towards Anakin Skywalker. Palpatine saw Maul as a pawn and used him to his advantage, but never considered him a direct threat to his life.
While talking to Qi'ra, Sidious made it clear that Maul was useless after he "died" on Naboo. When that happened, Sidious was ready to move on to a more subtle apprentice in Count Dooku. So, when Maul reappeared, Sidious had no use for him at all.
Summary. Palpatine despised Darth Maul for not being "quite good enough" and lost interest in him once his gaze turned towards Anakin Skywalker. Palpatine saw Maul as a pawn and used him to his advantage, but never considered him a direct threat to his life.
In the novelization he seems to feel… mild regret, maybe? He didn't expect Maul to die, and this might indicate a degree of respect for his apprentice and his skills; he didn't hold it against Maul that he lost, but honestly believed that the Jedi would be killed, so he might have felt that he underestimated them.
Regardless, based on the season 7 of The Clone Wars, and the fact that Darth Maul was aware of Anakin Skywalker's future and his role as the Chosen One, it is very likely that Darth Maul indeed feared Darth Vader to some extent, and preferred to avoid a direct confrontation with him, at least, on his own.
As Darth Sidious, Palpatine took Maul as an infant from his mother Talzin. While still apprenticed to his own master Darth Plagieus, Sidious trained the Nightbrother in the ways of the Sith. Maul was originally loyal to Sidious, obeying his every command without question.
By the time of Episode 3; Anakin is in his prime, and already one of the Jedi Order's most powerful “masters” (although technically still a knight), just behind Obi-Wan, Windu, and Yoda. Maul wouldn't stand a chance. Anakin stomps.
Maul had killed Qui-Gon and numerous other Jedi and started a huge Mandalorian conflict that left thousands dead. If he publicly announced that Palpatine was an evil Sith Lord, there would be few (if any) who would take him seriously.
Following Qui-Gon's death by Maul's hand on Naboo, Dooku visited Palpatine in distress about his apprentice's death, hoping to have one day made Jinn an ally while also directly stating he knew of Maul's existence.
Dooku despises Maul for being a threat to him, and for killing his beloved padawan Qui-Gon, and Maul hated Dooku for being his replacement. They also didn't regard each other as a true Sith: Dooku saw Maul as nothing more than a trained animal and Maul saw Dooku as a Sith pretender.
Even though Darth Plagueis knew of Maul's existence, he approved Palpatine training him since Maul was always meant to be cast aside. Nevertheless, Palpatine was setting Darth Maul up as his apprentice-to-be, knowing he would eventually kill Plagueis and take the Sith Master title for himself.
Of the three, Darth Vader was Palpatine's favorite "son." Indeed, once Anakin arrived on the scene, both Maul and Dooku became irrelevant to Palpatine's ultimate long-range goals.
Maul's skills, on the other hand, became stagnant as he spent the last few decades of his life keeping himself alive and plotting schemes. Given this, it makes sense Obi-Wan quickly defeated Maul in their final duel in Star Wars Rebels.
During Star Wars: The Phantom Menace, Darth Maul had no clue about Anakin and dismissed him as a child following the Jedi around. But with Maul's return during the Clone Wars, he likely found out about Anakin through his obsession with Obi-Wan.
Thought dead, Darth Maul survived his injuries by focusing on his hatred of Obi-Wan Kenobi, the Jedi who cut him in half. His shattered body was dumped amid the refuse of the junk planet Lotho Minor, where the once deadly warrior fell into madness, staying alive on a diet of vermin.
When he dies in Rebels you see Obi-wan holding the dying body of Maul. This is supposed to represent that he's gone beyond hatred or anger towards Maul, and instead feel pity that it had to end this way.
Thus, while Qui-Gon was less powerful than the likes of Yoda and Mace Windu, Palpatine feared him more. In the simplest terms, he was a wild card, and with the Sidious executing an intricate decades-long plan to seize power and destroy the Jedi, a wild card was something he just couldn't afford.
His acrobatic skills and prowess with a double-bladed lightsaber (which Andor imitated) instantly made him an icon. Yet his character was severely underdeveloped. With only 31 words to his name, Maul died as a salient assassin. However, that wasn't always the plan because his best dialogue never made it into the film.
