The move would allow the second-tier club, founded in 1885 and currently based in Richmond-upon-Thames in south-west London, to re-form as a semi-professional side.
"London Welsh has reached a difficult point in its illustrious history," chairman Gareth Hawkins said in a statement.
"Due to a playing budget of 1.7 million pounds and gates at games numbering as low as 400, the club’s current business model is totally unsustainable.
"The debts accrued from trading in this way have left the club with no alternative but to seek liquidation. Having to break that news to 40 staff members yesterday was extremely difficult," he added.
Hawkins said the board hoped the club could return to playing at its Old Deer Park home in the New Year once it had switched to a semi-professional model and a new company been formed.
It would also need to raise 300,000 pounds to regain a position within the RFU Championship.
"Richmond have proved that a club can compete in the Championship with a semi-pro model, and I firmly believe we can make this work in 2017," he said.
Founded by the Welsh community in London, the club provided a string of players to the great Wales and Lions teams of the early 1970s but fell on hard times in recent years.
In the Premiership as recently as 2015, with a base in Oxford, they had faced a series of winding-up orders over outstanding tax bills.
(Reporting by Alan Baldwin, editing by Ken Ferris)
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