The left-back, who turns 37 in December, joined the Major League Soccer side in January 2016 after seeing his deal with Roma terminated.
Galaxy have endured an underwhelming season and have won only three MLS games since the start of June, but Cole remains a key player and the ex-Arsenal and Chelsea star hopes to remain instead of returning to his homeland.
“I want to be here. We’re speaking. It’s in their court with me,” he told The Independent.
“There’s going to be a big change with the club and I think maybe it will happen next season.
“If I’m in those plans, we’ll find out. Who knows? I’m 37 so maybe they want to look a little younger.
“I’m just disappointed we’re losing, but I want to keep going until my legs actually give out. Thankfully they haven’t given up just yet.
“My legs feel okay. Maybe the head feels a little bit tired, but I’m enjoying football whether we win or lose. I love the game.
“I only want to stop when my heart for it finishes.
“It’s been a terrible season [for Galaxy]. Now we start again next season and hopefully do much better at home to change what we did this season.”
The American season is nearing its conclusion and the former England international is looking forward to his four-month break.
"[The close season] is long, but hopefully I’ll find a few gardens to clean, a few friends to take around the park!” he added.
“I’m happy I get to spend a little bit longer with my son and get ready for next season.”