Qatari banker Sheikh Jassim Bin Hamad Al Thani and British billionaire Jim Ratcliffe are both expected to increase their bids after the submission deadline was extended following confusion over the timing.
The Glazer family, United's current owners, have reportedly set a world-record valuation of £6 billion ($7.3 billion) for the club.
Sheikh Jassim's bid, promising to wipe United's $620 million debt and invest in a new stadium and training ground, in addition to backing for the men's and women's teams, remains confident that his offer is "the best for the club, fans and local community."
Meanwhile, Ratcliffe, who wants the 69 per cent stake owned by the Glazers, insists he will not pay a "stupid" price for one of football's most iconic clubs.
The Glazers, who have been deeply unpopular with supporters since their leveraged takeover in 2005, have invited external investment in November but could yet shun the option of selling a controlling stake in the club, with other parties interested in a minority shareholding.
Bidders are expected to hear from United next week, with another round of bidding still in play.
If one bid is vastly ahead of the others, it could be chosen to enter into a period of exclusivity, which would allow further negotiation ahead of a final sale.
If the Qatari bid is successful, it would give the Gulf state pride of place in the Premier League, the world's most-watched domestic competition, just months after hosting the 2022 World Cup.
However, it would also be controversial as Sheikh Jassim is the son of former Qatari prime minister Sheikh Hamad bin Jassim bin Jaber Al Thani, and his close links to the gulf state's ruling elite would raise questions over another Premier League club becoming a state-backed project.
United, three-time European champions, haven't won the Premier League since legendary boss Alex Ferguson led them to a 20th English title in his final season before retiring in 2013.
But they are enjoying a renaissance under Erik ten Hag's management this season and ended a six-year trophy drought by lifting the League Cup last month.
The Premier League has been urged by Amnesty International to tighten ownership rules to ensure they are "not an opportunity for more sport-swashing."