The 27-year-old Dubliner was stretchered off during a 1-1 draw at Ipswich in mid-January after damaging ankle ligaments and has been sidelined since.
Dawson has missed Barnsley's last nine matches after it was initially feared he had also broken his fibia, but is desperate to prove his fitness in time to be involved in the quarter-final clash at the Etihad Stadium.
"Daws is close," said Flitcroft, whose side stayed on the west side of the Pennines following Tuesday night's 1-1 draw at Burnley to continue preparations at the Dunkenhalgh Hotel and Spa near Blackburn.
"He wants to be and is gearing up to try and get this game in. We'll have a fitness check with him on Friday.
"I'd love Steve Dawson to be a part of Saturday. He epitomises everything I like in a human being.
"He carries fight, he carries hope, he carries spirit and I'd love him to be involved, but we might be a week short."
Flitcroft has no new injury or suspension worries. Rory Delap, on loan from Stoke, returned to the bench as an unused substitute at Burnley following a calf strain and is expected to be included, while fellow defender Jim McNulty is hoping to shake off an ankle injury.
Barnsley, who reached the semi-finals at Wembley in 2008 when they lost 1-0 to Cardiff, beat Burnley, Hull and MK Dons in previous rounds to book their date at the Etihad.
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