Burns, England's top scorer in the first Test, injured his left ankle while playing a football match and was taken for a scan.
Archer also did not train because of soreness in his right elbow and has also had a scan, according to a BBC report.
Pacer Mark Wood is yet to return to full fitness from his bout of sickness which has affected most in the squad while opener Dom Sibley is expected to be fit after being ill too.
"We've had some very frustrating moments on this tour so far but we've got some very talented players to pick from and we know whoever gets the opportunity will give us a great chance to win the game," England captain Joe Root told BBC Sport.
On Archer, he said: "We don't really want to go into a game with him not being 100 per cent and we also don't want to see him potentially miss a lot more cricket through playing him when he's not fit."
"All options are on the table at this point," Root said of his XI. "It's a great opportunity for them both to train well and put a case forward and then we'll make a decision."
( With inputs from IANS )