The 31-year-old joined from Middlesbrough in January and has since played five games, but has only completed the full 90 minutes once, in the 2-2 draw on Monday against Bournemouth.
The former Leeds United captain has been struggling with injury all season, but after featuring in three of Ipswich's last four games, he feels he deserves a full contract.
"I'd love to stay here to be quite honest, but it is not for me to decide - it is for the powers that be," he told the Ipswich Star.
"I hope I have shown enough in training and in the game (on Monday). I think I have.
"I am not really match-fit at the minute. My last 90 minutes was in December so the Bournemouth game was the first I've started for a very long time.
"I thought I did all right. In the second half I obviously tired, but the more games I play the better I will get."
Richardson also admitted that playing under Mick McCarthy is a big factor in wanting a contract, adding: "Mick is fantastic.
"He's honest and has this willingness to want you to work hard first and foremost. That is very important for a manager to get that across to his players. Along with Terry Connor, he does that very well.
"Mick's always up there when you talk about the best managers.
"When you talk about the manager of the season awards his name will always be bandied around in that category.
"He's a professional man, and hopefully he will stay and can take us up to the Premier League."