A spokesman for Ryan said that the deal had been completed and would be explained in detail at a press conference in London on Thursday.
Twenty-two-year-old Tomlinson is a global star as one member of five-piece boy band One Direction and is a lifelong fan of his hometown club.
He played for their reserve side against Barnsley earlier this year and had been close to a takeover before, but a move was twice delayed because of negotiations over image rights.
They have now been resolved and Ryan's spokesman told the Press Association: "It's done, the takeover is complete."
Ryan, who ended his 15-year association with Rovers in November, will be the lesser known of the two men but will doubtless be the more hands-on.
The businessman has extensive football experience and oversaw Doncaster's rise from non-league to the Championship as well as their move from Belle Vue to the Keepmoat Stadium before stepping aside in November to allow Terry Bramall and Dick Watson to take over.
The initial snag had related to Tomlinson's image rights, which are owned by his management team and would entitle them to a share of any money generated from the use of his image or name.
They believe that interest in the club will increase as a result of Tomlinson's ownership and it is understood they battled for a share of any extra revenue made through merchandise and extra ticket sales.
Tomlinson signed on as a player at the start of the season in a move to make money for the Bluebell Wood charity and turned out in front of nearly 10,000 fans for the reserve team in February.
By a strange quirk, Ryan has also played for Doncaster, with the then 52-year-old having brought himself on as a substitute in a 4-2 win against Hereford which came after his club had sealed promotion from the Conference in 2003.