The Red Devils head coach Erik ten Hag is said to be in the market for five more additions before the end of the summer window.
Real Madrid attacker Casemiro is likely to make the switch to Old Trafford in a £60m deal.
With Dean Henderson having been allowed to join Nottingham Forest on a season-long loan deal, Tom Heaton is the only second choice to David de Gea. Therefore, It is more likely that another alternative will be brought in as competition.
According to The Telegraph, Manchester United may approach Everton over a potential deal for Begovic over the coming days.
Having joined the Merseyside outfit on a free transfer in 2021, the Bosnia-Herzegovina international made a total of seven appearances during his first season with the Toffees.
The 35-year-old stopper has less than 12 months left to run on his contract.