On the back of Saturday's 4-0 defeat at the hands of Brentford, head coach Erik ten Hag is said to be keen to make five more additions before the end of the summer window.
If reports are to be believed, Real Madrid midfielder Casemiro is most likely to make the switch to Old Trafford in a £60m deal.
However, while other outfield positions need strengthening, Ten hag has a decision to make with regards to his goalkeeping department.
With Dean Henderson having been allowed to move to Nottingham Forest on a season-long loan deal, Tom Heaton is currently deputising as second choice to David de Gea.
Although Ten Hag may be content with keeping the veteran Englishman as backup, it is more likely that another alternative will be brought in as competition.
According to The Times, United may approach Everton regarding a deal for Begovic over the coming days.
Since being a regular at Bournemouth for a large part of his four-year spell between 2017 and 2021, the Bosnia-Herzegovina international has spent time on the substitutes' bench.
Having moved to Goodison Park on a free transfer in 2021, the 35-year-old made a total of seven appearances during his first season with the Toffees.
Three of those starts came in the Premier League, the most recent coming on the final day of the season after Everton have already avoided relegation to the Championship.
Head coach Frank Lampard will have little interest in losing the veteran's services unless he is able to find a like-for-like replacement over the next two weeks.
However, much may depend on whether Begovic specifically requests a transfer to Old Trafford, the stopper having less than 12 months remaining on his contract.
Although United have also been linked with Jurgen Sommer and Daniel Bachmann, they appear likely to remain at Borussia Monchengladbach and Watford respectively.