Midfielder Barkley has a toe injury while left back Baines has a back problem. Centre halves Jagielka and Distin have hamstring strains.
"Those four players are the ones we have got a question mark over," manager Roberto Martinez told the club's website (www.evertonfc.com) on Thursday. "Everyone else is fine."
Everton, seven points adrift of leaders Arsenal, are looking for their third home win in a row following victories over Southampton in the league and Championship side Queens Park Rangers in the FA Cup.
Norwich, who are sixth from bottom, have midfield pair Bradley Johnson and Wes Hoolihan and defender Sebastien Bassong available after injury.
"Our form away from home has improved from earlier in the season," said manager Chris Hughton. "That had to be the case and it needs to be the case again this weekend because we're going to a very good side."