They are the two main parties looking to take control of the club.
Club owners, the Glazer family, will now assess their position after the latest bid deadline passed at 22:00 BST on Friday.
Both bidders are eager for a resolution to a process that began in November.
BBC Sport has been told the offer from Sheikh Jassim is around £5bn and includes one figure that would go direct to the sellers.
It also includes a plan to make a significant amount of additional capital and infrastructure investment into the club.
It is understood the third bid deadline only applied to potential buyers of the whole club and not investment vehicles such as the US-based Elliott and Carlyle, both of whom are only willing to purchase a minority stake.
Finnish businessman Thomas Zilliacus says he lodged bids previously but would not submit a fresh offer.
The Glazers announced in November the board had “authorised a thorough evaluation of strategic alternatives”, including selling the club.
Investment bank the Raine Group was brought in to oversee the process after it handled the sale of Premier League rivals Chelsea.
After a deadline for expressions of interest in February and a second deadline last month, Friday marked a third deadline for bids.
It is understood large amounts of paperwork have had to be completed to get to the latest stage, leading all parties to believe the situation will be resolved before the transfer window opens at the end of the season.
There have also been suggestions this week that Ineos could restructure its offer, which initially was for the Glazer family’s combined 69% stake, in a way that could allow co-chairmen Joel and Avram Glazer to retain their personal shareholding.
It has been stressed Ratcliffe has been consistent in his view that he wants a majority stake, giving credence to the reports.
This move has already been attacked by the ‘1958’ fans group, who have demanded ‘Full Sale Only’.