The olive branch came a day after a Malaysian consortium won the right to redevelop London's historic Battersea Power Station site, beating a rival bid from Chelsea to build a new 60,000-seater stadium at the site.
Champions League winners Chelsea, owned by Russian oligarch Roman Abramovich, have played at Stamford Bridge since 1905. However, the stadium has a capacity of just under 42,000 - small by comparison with many of the club's European rivals.
"It now looks highly unlikely that CFC [Chelsea Football Club] will be moving away from their historic home," said Hammersmith & Fulham Council leader Nicholas Botterill.
"The council remains committed to working with the club to explore all possible avenues for keeping the Blues here at their original home," he added.
"The door is open and the council is again ready to sit down and discuss how Stamford Bridge can be appropriately upgraded so that it is fit for one of Europe's leading football clubs."
Chelsea Chief Executive Ron Gourlay told Reuters last month that the club's preferred option was to stay and expand the stadium but that did not appear economically viable and would be complicated by planning issues.
Malaysia's Sp Setia Bhd and the property arm of Sime Darby Bhd were named as preferred bidder on Thursday to develop the derelict Battersea site on the banks of the river Thames in a 400 million-pound deal.
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