Sheikh Jassim is in a bidding war with British billionaire Jim Ratcliffe after the pair emerged as the main contenders to buy the Premier League club.
United announced in November that the board was exploring “strategic alternatives to enhance the club’s growth”, with a full sale one of the options.
Current owners, the Glazer family, are understood to value United at £6 billion ($7.5 billion) and the Raine Group was brought in to oversee the process.
The fifth offer remains for 100 percent of the club and Sheikh Jassim’s bid team have set Friday as a deadline to make headway in discussions, reports said on Wednesday.
While the offer is understood to remain after that point, the end of the week would mark the end of their ongoing engagement in the process.
INEOS chemical company founder Ratcliffe appeared to be the front-runner after a third round of bidding closed in late April.
Ratcliffe, a boyhood United fan who failed in his attempt to buy Chelsea last year, reportedly wants to purchase a controlling stake of more than 50 per cent.
That would allow the Glazers to remain involved — a move that would be deeply unpopular with supporters.
United lost Saturday’s FA Cup final to Manchester City but they won the League Cup and finished third in the Premier League in Erik ten Hag’s first season in charge.
The lengthy ownership saga means there is an air of uncertainty over the future of the club as they prepare for an important summer transfer window.
Co-owner Avram Glazer was at Wembley for Saturday’s 2-1 defeat to City and ignored questions about the takeover as he left through the interview area.
Asked about the potential takeover and summer improvements, United boss Ten Hag said: “I think in this moment I don’t want to discuss that.
“It’s about finalising the season. We have to be quiet, analyse the season, go into the depth, then set the right conclusions and then take action.”