Egan suffered a clash of heads with Andreas Cornelius during the Republic of Ireland's 1-1 draw with Denmark on Monday, and was replaced at half-time.
It makes him a doubt for Sunday's televised game at Bramall Lane and threatens further disruption to a defensive unit which will already be without goalkeeper Dean Henderson. The experienced Phil Jagielka is on standby to make his first Premier League start for the Blades in 12 years.
“He has a chance - we’re going to leave it to the last minute but we’ve got replacements if needed,” said manager Chris Wilder of Egan. “If John isn’t fit Phil will slot in there and his experience in big games will be huge.”
Egan played another 20 minutes before being substituted on Monday but given the nature of the injury, the Blades will have to be careful with him.
Monday's drawn condemned Mick McCarthy's Republic of Ireland to the Euro 2020 play-offs, where they could face Northern Ireland in a one-off game to decide who plays in the tournament proper.
The Republic are in Slovakia on March 26, when Northern Ireland travel to Bosnia-Herzegovina. The winners of the first game will travel to the winners of the second five days later to determine who plays in next year's itinerant competition, which will feature four games in Dublin.
Egan, Henderson, and the rest of Sheffield United's five-man defence have yet to miss a Premier League game this season. Central midfielder John Fleck is the only player behind the front two not to have started every league match this season, having missed two games through injury in August.
Henderson is ineligible because he is on loan from the Red Devils. His place will be taken by Simon Moore, who will be making his Premier League debut at the age of 29.
Centre-back Jagielka is in his second spell at Bramall Lane, having left for Everton in 2007. The Sale-born 37-year-old started four top-flight games for the Toffees last season, the last of them in April, and has made two League Cup starts and three appearances from the bench since rejoining the Blades in the summer.
In total he has made more than 600 senior appearances, as well as winning 40 England caps.
Manchester United will also be without a Premier League ever-present on Sunday, with Scott McTominay out with an ankle injury. Fellow midfielder Paul Pogba is not ready to return after his own ankle trouble.