The council released a statement on Tuesday morning reiterating their desire for the Blues to stay put in the wake of Roman Abramovich's failed bid to purchase the freehold of the club's current home last Thursday, something that was seen as a precursor to a move to a new 60,000-seater stadium.
Chelsea supremo Abramovich was unable to convince supporter-led group Chelsea Pitch Owners - who the club sold the land beneath Stamford Bridge to in the 1990s to stave off property developers - to hand back the freehold and give the Blues the green light to relocate.
Chelsea, who have yet to decide whether to move, have been frustrated in their efforts to expand Stamford Bridge by a meaningful capacity at a cost that makes economic sense.
They would consider relocating to Earls Court or White City, which would keep them in the borough, while Battersea is a possible destination outside Hammersmith and Fulham.
The council, who recently agreed to grant Fulham planning permission to enlarge Craven Cottage, said in a statement: "In light of the recent decision by Chelsea Pitch Owners plc to retain its freehold of the ground, we reaffirm our desire to see Chelsea FC remain in the borough and our commitment to help the club to remain at its historic home of Stamford Bridge.
"We recognise that all our clubs will need greater capacity to accommodate growth in their fan bases and that new Financial Fair Play rules necessitate generating extra match-day revenue to remain competitive with the leading English and European teams.
"The council now proposes to examine whether there are planning options to expand Stamford Bridge to accommodate a larger capacity, recognising that such a project must be economically viable, benefit local businesses and not unreasonably affect residents.
"We look forward to working closely with Chelsea FC and all local stakeholders as soon as possible to examine the options of redeveloping Stamford Bridge."
A Chelsea spokesman said in response: "Chelsea Football Club welcomes the London Borough of Hammersmith and Fulham's commitment to see the club stay in the borough and importantly their recognition of our need for greater capacity at Stamford Bridge.
"As an initial step in a renewed dialogue with the borough, we would like to explore promptly with LBHF how they propose helping us address the club's current problem of Stamford Bridge having substantially reduced capacity for big games.
"We will be happy to meet borough officials to discuss this further and especially their desire to see us stay at Stamford Bridge."
The 'Say No CPO' campaign, which spearheaded opposition to Abramovich's bid last week, were "delighted" with the council's intervention, urging them and Chelsea to "exhaust all possibilities" that might allow the club to stay put.
They added: "We're sure that most fans will welcome the chance to see Chelsea Football Club remain at their historic home."