The Norwegian full-back has been dropped into the free agent pool at the end of his contract at Crystal Palace.
The 25-year-old is known to be keen on securing another English club after three years at Selhurst Park.
Bolton boss Dougie Freedman, the man who took him to Palace in 2011, has expressed a desire to be reunited with Parr at the Reebok Stadium.
It is, however, Ipswich who have held positive talks regarding a possible deal and would appear to have edged to the front of the queue.
“If Jonathan wants to go there, there is a big chance it will happen,” Roger Parr, Jonathan’s father and advisor, told Aftenposten.
“At Ipswich, he will be first choice. Jonathan has had a meeting with the manager (Mick McCarthy) and they had a positive talk.
“We know that it is a really good club with a nice stadium and good supporters. That is important.
“Everything looks positive.”