The 31-year-old, who began his career at Portman Road in 1998, is a free agent after he was released by Sunderland at the end of last season.
The former Newcastle defender trained with West Ham earlier this summer to maintain his fitness before later linking up with the Ipswich squad.
He played in Town's 2-1 friendly win over Gillingham on Tuesday night and boss Mick McCarthy later hinted Bramble could rejoin the club on a permanent basis.
When asked about a potential return to the Tractor Boys, Bramble confirmed it was a possibility.
"I'm not ruling anything out, there are a few options," he told the Green 'Un. "The main is for me to stay fit ahead of wherever I may end up.
"I've always said that one day I will come back and play for Ipswich again. Whether it's now or in a couple of years' time, who knows?
"Ipswich is in my heart. I'm born and bred from the town, so it means everything to me. I've always looked out for their results ever since I left and it's been disappointing to see them struggle at times because I think they should be a Premiership club.
"They seem to be picking up under Mick McCarthy though and I see no reasons why they won't be pushing next season."