The 20-year-old will travel to United's Austrian training camp after signing a three-year contract, which is subject to international clearance.
Mentel will provide back-up to compatriot Dusan Pernis, who made an impressive start to his United career after making his debut in January.
United boss Peter Houston told the club's official website: "Filip is a highly promising young keeper. That he's been at Manchester City for the last few seasons confirms his credentials, but obviously he's found it difficult to break through there given their presence in the transfer market these days.
"We had him at Tannadice for a few weeks last season and, while he impressed us all greatly, we didn't think we would get the opportunity to sign him.
"I'm therefore delighted that this opportunity has now arisen and we've been able to secure him for Dundee United.
"Filip will prove a great back-up to Dusan Pernis and Stevie Banks and we also expect to see him challenge both of them, in time, for the first-team jersey.
"Filip is another player we've signed with the future in mind and with young Marc McCallum, who is only 17 but features in the Scotland Under-19 squad, we now have a very strong goalkeeping situation at the club."