Dutch striker Jimmy Floyd Hasselbaink is set to leave Middlesbrough after his contract with the club expired.
Boro has decided not to give the 34-year-old a new contract, and Celtic is interested in signing the Dutchman.
Southgate told Boro’s official website: “I have spoken to Jimmy and explained to him that we are not going to renew his contract.
“It was a very difficult decision to make because he has done an excellent job for us.
“I have to make some tough decisions in this job and this one certainly wasn’t easy, but I have to do what I think is right for the club.
“Jimmy’s a good player and I’m sure he will score goals for somebody. But I’ve got excellent striking options here and feel we really need to strengthen in other positions that have to take priority.
“He scored some very important goals for us over the last two seasons, helping to get us into Europe in the first place and then playing a big part in our run to the UEFA Cup final.”