Aston Villa had hoped to keep the Belgium midfielder and offered him new terms, but the release clause in his existing contract gave Manchester United the ability to move regardless.
Tielemans, 29, reached the World Cup quarter-finals with Belgium this summer but missed their 2-1 defeat to Spain through injury. He has spent three seasons at Villa Park since arriving on a free transfer, and previously spent four and a half years at Leicester City, including a loan spell there.
The move comes during a busy period of business for Villa.
The Daily Mail reported on Monday that left-back Lucas Digne is a target for Paris Saint-Germain under a similar release-clause arrangement, while Villa are separately closing in on a club-record deal for Switzerland international Johan Manzambi, expected to cost close to £60m.
Villa are operating under strict UEFA financial sustainability rules that prevent them spending more on football costs than they generate in income, meaning any deal for Manzambi would need to be offset by departures elsewhere.
The exits of both Digne and Tielemans, who each had two years left on their contracts, would satisfy that requirement, with Tielemans' sale counting as pure profit given he joined on a free transfer.
Losing the midfielder would still be a blow for manager Unai Emery, for whom Tielemans has been a key player. Villa's central midfield options are already stretched, with Tielemans' Belgium team-mate Amadou Onana out for an extended period after rupturing his ACL at the World Cup, and Boubacar Kamara still working his way back from a knee injury sustained in January. Emery had planned to focus the summer on strengthening his attacking options, but may now need to prioritise midfield instead.
For United, the move is part of a wider rebuild in central midfield. The club has also agreed a £50m deal for Chelsea's Andrey Santos, who impressed on loan at Strasbourg last season, as manager Michael Carrick looks to replace Casemiro following his departure after four years at Old Trafford. United are additionally without Manuel Ugarte, who suffered a long-term knee injury on World Cup duty with Uruguay that he described as "the most serious injury a footballer can face."
It has been a difficult summer for United in the market. The club missed out on its two primary midfield targets, Elliot Anderson and Mateus Fernandes, who joined Manchester City for £116m and Tottenham for £85m respectively, and pulled out of a move for Atalanta's Ederson on medical grounds. Tielemans, by comparison, represents a considerably cheaper option with substantial Premier League and European experience.