Of course they do. The Bosnian ankle-biter has been absolutely superb for Tony Pulis's side since he moved to the Riverside on loan from Everton in the January transfer window.
Boro have won four of the six games in which Besic, a dynamic defensive midfielder, has started and he has wasted no time making himself a bona fide favourite among fans and club legends alike.
And the Northern Echo claims that Pulis could be in luck. Everton are open to the idea of the 25-year-old leaving on a permanent basis at the end of this season, although they want more than the £4.5 million they paid Ferencvaros back in the summer of 2014 after Besic stood out at the Brazil World Cup.
But with Besic in excellent form, surely it would be too soon for the Toffees to cash in. Especially as Morgan Schneiderlin - another midfielder who would fetch far more in the transfer market - struggling so badly at Goodison Park.
The Frenchman has been a big disappointment since his £24 million move from Manchester United but, unlike Besic, he cannot put this down to a series of injuries. Schneiderlin has featured regularly but has been nowhere near the level that earned him such high praise at Southampton.
Besic, by comparison, has not really had his chance due to a long list of fitness issues, so Everton should keep a man on the rise and sell another whose decline is showing no signs of stopping.