Monk struck a 90th-minute winner in Tuesday night's Capital One Cup victory over Crawley Town in his first appearance of the season after recovering from a back injury. Bristol City made an approach for the 33-year-old, who is not in Laudrup's first-team plans, during the summer but the transfer failed to materialise.
"I've been injured for the past five weeks but there was speculation before that about a loan to here, there and everywhere," he said.
"I've been training for a week, played my first game and I'm going to build on that and see where it takes me.
"I've said all along I don't want to leave. Hopefully I can force my way into the manager's plans in some capacity in the near future. If not, I'll assess it again.
"At the moment I just want to get fit and everything depends how good I feel in the next couple of weeks. If I get a couple of weeks' training under my belt and it looks like I've got absolutely no chance, I'll have to have a look at it.
"January is the realistic time to move if I'm not involved at all, but hopefully I will be involved. I'll keep pushing and pushing, but I'm realistic as well. I'm not saying I should be starting Saturday or anything stupid like that."
Laudrup has retained Monk as club captain in recognition of his eight years of service to Swansea, despite his uncertain future at the Liberty Stadium.
"That means a lot to me - this is my seventh season as captain," Monk said.
"I've always been proud to be captain of this team and proud of what we've achieved in reaching this level. But I want to play and be out with the boys, helping the team."
Source: PA
Source: PA