Scotland defender Charlie Mulgrew [left]
Middlesbrough are said to be looking at central defenders as they anticipate Ben Gibson's departure, while Hull sold Curtis Davies to Derby County earlier this week.
Middlesbrough defender Ben Gibson
And Mulgrew has told the paper he is keen to remain in the second tier next season: "I don't know what will happen next but I'd be happy to stay in the Championship if that's what transpires.
"I feel it's a league that suited me and I've felt comfortable in it."
Mulgrew, who had a two-year spell at Wolverhampton Wanderers a decade ago, returned to English football last summer from Celtic.
Former Celtic defender Charlie Mulgrew
The 31-year-old made 28 Championship appearances in his first season at Blackburn, scoring three goals and creating another.
At the time of writing, Middlesbrough are expected to confirm Garry Monk as their new manager, with Leonid Slutsky taking the Hull reins earlier today [Friday].
Leeds United manager Garry Monk
And it remains to be seen how those appointments will affect the pair's alleged interest in Mulgrew, who can also play at left-back and central midfield and has 26 caps for Scotland.
Should Middlesbroguh/Hull City move for Blackburn's Mulgrew this summer?