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.

David Cameron also nominated Tory and Labour members, including former attorney general Dominic Grieve.
Mr Robertson was among a number of MPs who had called on the committee to investigate an RAF drone strike in which two British citizens were killed.
He is expected to use his position on the committee to re-affirm those calls for an investigation.
It was confirmed on Monday by the UK government that Cardiff-born Reyaad Khan, 21, and Ruhul Amin, from Aberdeen, died in Syria, alongside another fighter, in the first targeted UK drone attack on a British citizen.
The men had gone to the conflict zone to fight for the so-called Islamic State group.
Defence Secretary Michael Fallon said the drone strike was a "perfectly legal act of self defence".
He added that the UK would not hesitate to launch more secret drone strikes in Syria to thwart terror plots.
The new appointments to the intelligence and security committee, which can summon intelligence chiefs to give evidence, include;
London Welsh are "extremely disappointed" by the Rugby Football Union's decision to expel them from the Championship and the professional game.
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The prime minister has appointed SNP MP Angus Robertson to Westminster's intelligence and security committee.