Arsenal moved up to third with as a 4-0 win over Swansea City saw them move ahead of Liverpool on goal difference.
Two own goals from strugglers help Gunners to comfortable victory.
Costa, 28, trained alone at Cobham for three days and did not travel to Leicester City with the rest of the squad.
Spurs moved up to second in the table as Harry Kane's hat trick helped them see off the Baggies 4-0, but lost defender Vertonghen after 64 minutes.
Jose Mourinho's side head into the match on a nine-match winning run in which Mkhitaryan has been key.
Tottenham are in ominous form; their sixth straight Premier League win a 4-0 stroll over West Brom that kept up their title challenge momentum.