The Championship duo are both in need of a first-choice goalkeeper ahead of the new season and, according to page 68 of Sunday 3 June's Daily Mirror (via HITC), both clubs have identified Ward as their first choice.
Norwich are be looking to replace Manchester City loanee Angus Gunn, who has returned to his parent club after a string of impressive performances in the Championship last term.
Meanwhile, Swansea goalkeeper Łukasz Fabiański has been heavily linked with a move away from south Wales, although the club may wait until they appoint a new manager before deciding whether or not to progress with a deal for Ward.
Ward could prove to be a good signing for either side, as the 24-year-old has previous experience of achieving promotion to the Championship.
He was a key figure in the Huddersfield side that got promoted in 2016-17 season, catching the eye of Premier League scouts, but was unable to earn another move upon his return to Liverpool last summer.
Norwich's interest could be down to the club's sporting director, Stuart Webber, who was at Huddersfield during Ward's stay and played a key part in signing the goalkeeper.
The Canaries have also been linked with a loan move for Manchester United goalkeeper Dean Henderson, who has attracted interest following a successful loan spell in League One with Shrewsbury Town last season. Henderson caught the eye after a previous impressive loan at Grimsby Town during the 2016-17 season.