McKenna worked under Solskjaer at Manchester United alongside Michael Carrick between 2018 and 2021, with the coaching staff guiding the club to a runner-up finish in the 2020-21 season.
McKenna remained at United following Solskjaer's sacking before leaving to take charge of Ipswich in December 2021.
The Northern Irishman led the club to three promotions during his time in charge, taking Ipswich from League One to the Premier League, but recently stepped down to take a break from management.
According to the BBC, Solskjaer has emerged as a serious candidate to succeed McKenna, having been seeking a return to management since his exit from Besiktas last August.
The Norwegian had a brief and difficult spell in Turkey, lasting only a matter of months before his dismissal, and has since worked as a technical observer for UEFA while waiting for the right opportunity to return to club management.
Solskjaer was also considered a contender to return to Manchester United on an interim basis earlier this year following Ruben Amorim's departure, before the club opted for Carrick instead.
That United connection, along with his existing relationship with McKenna, is understood to be a factor in his consideration for the Ipswich role.
Gary O'Neil has also been described as a leading candidate for the Ipswich job.
The 43-year-old has previously managed Bournemouth and Wolves in the Premier League, and is currently in charge of Strasbourg, having succeeded Chelsea-bound Liam Rosenior in January.
Strasbourg finished eighth in Ligue 1 under O'Neil, missing out on European qualification by a single point.
O'Neil remains under contract at Strasbourg for a further three years, which could complicate any move back to English football this summer.