Both clubs want a new defender this summer to solve weaknesses which were highlighted last season.
Kounde is also wanted by Arsenal but Mikel Arteta’s side instead appear set to sign Brighton’s Ben White in a huge £50million transfer.
None of Tottenham’s centre-backs sufficiently convinced last season that they can be a regular feature in their back line.
The performances of Davinson Sanchez and Eric Dier came under particular scrutiny.
United are also believed to be interested in Real Madrid’s Raphael Varane and Villarreal’s Pau Torres.
The Red Devils’ interest in other targets might allow Tottenham a free run at Kounde.
Spurs are “highly appreciative” of the 22-year-old, who could leave for less than his £68m release clause, according to journalist Fabrizio Romano.
The north Londoners, who are still yet to appoint a manager, are targeting two defenders this summer.
Former Fulham loanee Joachim Andersen is also wanted.
Meanwhile, former Wolves boss Nuno Espirito Santo remains favourite to become the new Tottenham head coach but the farcical search for Jose Mourinho’s successor continues.
Mourinho was dismissed in April and Spurs have so far reportedly come close to appointing each of Antonio Conte, Paulo Fonseca and Gennaro Gattuso but talks have failed at the final hurdles.
Fabio Paratici, Spurs’ new managing director, wanted to appoint Nuno during his time at Juventus, Romano claims.
Rangers manager Steven Gerrard remains a target alongside Brighton’s Graham Potter.
Nuno, Gerrard and Potter all fit the profile of Levy’s preferred head coach, which he described in a statement to supporters.
“We shall focus on the recruitment of a new Head Coach,” Levy wrote to fans after announcing the formation of a Club Advisory Board last month.
“We are acutely aware of the need to select someone whose values reflect those of our great Club and return to playing football with the style for which we are known – free-flowing, attacking and entertaining – whilst continuing to embrace our desire to see young players flourish from our Academy alongside experienced talent.