Easter linked up with Brendan Rodgers' team after moving from MK Dons earlier this month and marked his first start for the club with the winner in Tuesday's 1-0 victory over Barnsley.
The 28-year-old is keen to prove himself further in south Wales, though, and is open to remaining at the Liberty Stadium beyond the current length of his existing deal.
"I told the manager from day one that I'd like to be a Swansea player permanently," Easter told the club's official website.
"Hopefully Tuesday helped the case and hopefully I can go on and start a few more games and show what else I can do. It's great to be back home. I'm close to my family and back with my wife so it's great.
"There are some really great players here and a great manager. I feel a part of it and it's great to be here."
Source: PA
Source: PA