London Irish, who have been involved in protracted takeover talks with an American consortium, last month failed to pay their employees on time according to reports in British media.
The RFU said London Irish's proposed takeover had led to "speculation about the future of the club", highlighting the impact of the uncertainty on players, staff, and fans.
"The RFU has therefore set a deadline of May 30th by which either a takeover of the club has been completed and approved by the RFU...." the governing body said in a statement.
"Or, the club evidences that it will continue to be funded to operate throughout the 2023-24 season..."
The RFU added that it had set the deadline to "avoid a scenario where the club enters insolvency mid-season."
Two other clubs, Worcester Warriors and Wasps, have been affected by financial issues this season and were removed from the Premiership after going into administration. Wasps will play in the second-tier Championship next season, but Worcester will not.
London Irish ended in fifth place in the Premiership standings.
