Bramble has trained with the Tractor Boys and played in their 2-1 friendly win over Gillingham on Tuesday.
Now, ex-Mackem McCarthy could be ready to offer the 31-year-old a contract - but Bramble would prefer a Premiership club.
Mick McCarthy: “I only met up with Titus for the first time at our pre-match meal.
“He asked if he could come in for training and wanted to play.
“He’s obviously interested in coming here and we are interested in him, otherwise he wouldn’t be playing for us.
“He’s shown a good commitment to come and play but it’s early days yet. No more than that.”
Bramble was a high-earner at the Stadium Of Shite and he will have to drop his wages by THREE QUARTERS to sign for Ipswich.