"It has become clear to me that the Board is intent on selling the club to New England Sports Ventures (NESV) to the exclusion of all other parties, regardless of the merits of their bids," Lim said in a statement.
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"In these circumstances, I am not able to proceed with my intention to acquire the club
A London High Court ruling on Wednesday gave the Liverpool board the right to proceed with a 300 million pounds sale of the club to NESV, who also own the Boston Red Sox baseball team.
However, current owners George Gillett and Tom Hicks, who lost the case, later gained a temporary restraining order on the sale in a Texas court.
The takeover case returned to London's High Court on Thursday as the board seeks to overcome the restraining order.
Lim, who made his fortune through stockbroking, said he had received no response to his offer.
"I have tried to engage constructively with the board and Royal Bank of Scotland based on an offer, funded from my existing resources, providing greater value for Liverpool Football Club, more cash for players, full repayment of all bank debts and a long term personal commitment to build a better future for the Club and its supporters," he said.
"The board and RBS have chosen not to respond or to discuss my offer with me. My representatives even offered to meet the board last night. This was ignored, although NESV was invited to attend that meeting."
Lim said his interest in buying Liverpool could be revived if "current events cause the circumstances to change."
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