Boro boss Chris Wilder wants to add attackers to his squad after seeing Andraz Sporar, Folarin Balogun and Aaron Connolly all depart the club.
He has his eye on Southampton’s Armstrong and Newcastle’s Gayle, however a move for either player is rated as being some way off.
Saints snapped up Armstrong from Blackburn Rovers last summer for a fee in the region of £15m, but he has struggled to make his presence felt in the Premier League.
The 25-year-old scored just twice in 23 Premier League outings in the recently concluded season.
Gayle meanwhile is well down the pecking order at Newcastle and the Magpies are expected to add further strikers this summer.
The 33-year-old clocked just 30 minutes of football in the Premier League for Newcastle over the course of the 2021/22 campaign.
He still has another two years left to run on his contract at St James’ Park.
