The Glazers revealed in November last year that they were prepared to explore outside investment or a full sale of the 20-time English champions.
Following the first round of offers, Ratcliffe's main competition for Man United appears to be Sheikh Jassim bin Hamad Al Thani.
Ratcliffe, who already owns Nice and Lausanne Sport, is only interested in buying the Glazers' shares, which would give him a controlling stake, and he has promised not to add any more debt to the club.
Sheikh Jassim, meanwhile, wants to purchase 100 per cent of Man United and is prepared to wipe off their existing debt of around £515m.
Ratcliffe visited Old Trafford last week to discuss a potential takeover before he visited Carrington where he briefly met Man United boss Erik ten Hag.
In contrast, Sheikh Jassim did not attend talks in person, instead opting to send a team of representatives on his behalf.
Ratcliffe and Sheikh Jassim allegedly submitted opening bids in the region of £4bn and £4.5bn, falling below the Glazers' minimum valuation of £5bn.
The Raine Group, who are handling the sale on the Glazers' behalf, have invited prospective buyers to table their second bids by Wednesday.
Ahead of a potential new offer, Ratcliffe has revealed that he is not willing to pay a "stupid" price to buy Man United when he was asked about a takeover bid in a recent interview.
The INEOS owner told the Wall Street Journal: "How do you decide the price of a painting? How do you decide the price of a house? It's not related to how much it cost to build or how much it cost to paint.
"What you don't want to do is pay stupid prices for things because then you regret it subsequently."
The 70-year-old's stance differs from Sheikh Jassim, who is reportedly prepared to overpay to become Man United's new owner.
Ratcliffe has promised to position the Red Devils as a "community asset" and insisted that his interest "would be purely in winning things".