Green has made four appearances for Ipswich since joining on loan from Leeds United last month and his contract at Elland Road expires at the end of the season.
The 30-year-old is enjoying his time at Portman Road and is hoping to be offered the chance to remain at Ipswich for the long term.
"I've loved it here from day one and it would be great to stay, especially if we can win promotion," said Green.
A 1-0 home win over Birmingham on Saturday left Ipswich four points adrift of the play-offs and Green feels Mick McCarthy's side are in the fight for a top-six finish.
The Republic of Ireland international added on the club's official website: "It's a massive win for us and it's put us back into the mix.
"We're four points off the play-off places and it's important we go to Middlesbrough and get something now.
"I thought we were on the front foot right from the first whistle and we deserved the three points.
"We got our goal and dug in against a very good Birmingham side, so we're delighted to get the three points."