Arsenal's Mesut Ozil celebrates scoring his side's second goal of the game during the Premier League match at the Riverside Stadium, Middlesbrough.
Blues boss Antonio Conte is in the market for a long-term replacement for John Terry and Gibson is certainly in the same mold as the long-serving Chelsea skipper so he'd be an excellent addition as the west Londoners prepare for their return to the Champions League.
The Teesside outfit find themselves six points from safety with just six games of the season remaining having extended their winless run to 15 matches with a 2-1 defeat at the hands of Arsenal on Monday night.
He said: "If you asked the other teams, 'Are Middlesbrough one of the three worst teams in the league?', I don't suppose many would say we are".
"You have to have this positivity to keep going, otherwise you won't win anything". If we don't, it will be goodnight. We're convinced one win will change it all.
Alexis Sanchez's expertly-dispatched 42nd-minute free-kick had given the visitors the lead at the end of a tight first half, but Alvaro Negredo dragged Boro back into it five minutes after the restart and had Petr Cech not kept out Daniel Ayala's close-range header 11 minutes later, the outcome might have been very different.
Clayton said: "It is hard to come out of games like that".
Gibson only signed a new long-term contract last summer but Boro know they'll have no chance of keeping hold of him if they go down while a move could still be on the cards even if the Teeside club manage to avoid the drop. But we will not go down because we have been unlucky, we will go down because we haven't been good enough - if we do.