The Norwegian defender has been at the club since January, 2010, when he signed on for three-and-a-half years.
However, he has frequently been affected by injuries and has made only 14 appearances this season, slipping behind Efe Ambrose, Kelvin Wilson, Charlie Mulgrew and even Mikael Lustig in the central defensive pecking order.
Neil Lennon does not want to lose the 22-year-old but admitted yesterday that the deal which has been rejected by the Norwegian may be the last that is offered.
"Rogne has been offered a contract but it has been refused," the manager said. "We may go back to him with another offer but [there are] no guarantees on that. We don't want it to fester on; we want a result one way or the other. Thomas is a good player and we would like to keep him around. He has had to be patient because Efe Ambrose, Charlie Mulgrew and Kelvin Wilson have done exceptionally well, but whenever Thomas has come in he has never let us down."
Far more significantly for Celtic, a similar blank has been drawn so far in terms of reaching an agreement with Gary Hooper, who now seems certain to leave for a significant transfer fee during the close season. The striker has recently changed his agent but Lennon suspects Hooper's camp do not intend to make any announcements about his future until the end of the season. The contract offer currently on the table from Celtic is likely to be withdrawn before then.
"The contract is still there," Lennon said. "Eventually we will decide to pull it if we are not going to get any encouragement that he is going to sign it. The ball is entirely in their court.
"They have not intimated to me that he is going to sign a new contract. I'm assuming they will look at things in the summer. I don't know how long the contract will be on the table. You'd need to ask Peter [Lawwell, Celtic's chief executive] that."
Contract talks will also be held over the next couple of months with Paddy McCourt, whose current deal ends this summer. Meanwhile, Celtic are available at a prohibitive 1/9 to win at home to Dundee in the Clydesdale Bank Premier League tomorrow. With the hosts having scored 18 times in their last four home league games, the managerless visiting team will enter Parkhead with some trepidation.
Lennon admitted having sympathy for Barry Smith, dismissed this week by the club currently 15 points adrift at the bottom of the league.
"From the outside it looks very harsh – the timing of it seems a bit strange," Lennon said. "They seem to be cast adrift at the minute. It is going to take a monumental effort to turn things around and I hope it does not turn round at our expense on Sunday.
"He [Smith] had his hands tied at the start of the season because he was not sure which division he would be playing in [because of uncertainty over Rangers' status]. It has been a struggle for them but it was always going to be. It is a big step up from the first division to the Premier League.
"There has been [managerial] stability in Scotland [this season] which is quite pleasing. I just find the timing of it a bit funny. You're looking at a full season, maybe 18 months, before it can be considered a manager's team so it's possibly two years before you can judge that. We get judged in the job in five minutes. It's the knee-jerk capital of football, up here."
Lustig will miss the next two or three weeks with bruising to a bone while Celtic will learn at the start of next week whether Scott Brown will require surgery on his abductor muscle.
"I'm speculating that he might need surgery but we'll see what the surgeon says," Lennon said. "An operation like that is normally four to six weeks [recovery time] but we are expecting results back on Monday or Tuesday."
Mulgrew will miss today's match because of a muscle strain but could return to face Motherwell at Fir Park on Wednesday. After hosting Dundee tomorrow Celtic have a run of four consecutive away games against Motherwell, St Mirren, Juventus and Ross County.