The RFU announced on Tuesday that the Exiles had not met the governing body's criteria to be granted a new licence.
However, Welsh believe they had satisfied the requirements.
The club, who went into liquidation last month, say they will "deal with the ramifications" of the announcement before deciding whether to respond.
A phoenix entity, named Rugby 1885 Limited, had been granted a temporary licence on 22 December to play their last two league games after Welsh went into liquidation, although the club were deducted 20 points.
On expelling the club from the second tier, the RFU stated the Old Deer Park side were unable to provide a bond or show evidence they would be able to pay rugby creditors in full.
Welsh said they had worked "very closely" with RFU chief executive Ian Ritchie, director of professional rugby Nigel Melville and legal and governance director Angus Bujalski in order to to meet the conditions.
The Richmond-based club were a Premiership side as recently as 2015, when they were based at the Kassam Stadium in Oxford but were relegated after losing all 22 games.
Welsh have suffered financial difficulties since then and had hoped to secure investment from a United States-based group, but that deal collapsed in November.
Exiles coach Sonny Parker criticised the RFU of acting "through self-interest" and announcing the decision with "no regard to those impacted".
The former Wales international, who is backs coach and team manager, said the decision to remove them from the Championship was made public before players and coaches were informed.
"I can live with no licence but not allowing us time to speak to players and staff is unforgivable," the 39-year-old said in a statement on Twitter.
London Welsh are "extremely disappointed" by the Rugby Football Union's decision to expel them from the Championship and the professional game.