The Exiles won this season's Championship after beating Cornish Pirates in both legs of the final.
But the Richmond-based club were told just hours before the first leg kicked off in Cornwall they did not meet minimum standards criteria set down by English rugby's Professional Game Board for Premiership entry.
London Welsh played the final's second leg at the Kassam Stadium in Oxford, which is thought to be their preferred venue should they gain top-flight status.
As things stand, Newcastle will remain in the Premiership next term despite finishing bottom by a point behind Wasps this season.
But should London Welsh succeed in overturning an original decision that went against them, then they will go up and the Falcons be relegated.
In a statement, the RFU said: "The Rugby Football Union has today (Thursday) received London Welsh's appeal against the decision that the club failed to meet the minimum standards criteria set out by the Professional Game Board for promotion to the Aviva Premiership.
"It is proposed that the appeal hearing, which will take place before an independent panel, will be held on June 21 at the London Bloomsbury Hotel.
"An expedited timetable has been agreed with London Welsh, with the proposed date of June 21 the earliest possible time to allow for the exchange of cases and evidence.
"During the appeal, no further comment will be made."