Stockdale, who played five games on loan for the Tigers at the end of last year, is being eyed as a possible long-term loan signing.
Bruce has made recruiting a new goalkeeper a top priority as Hull step up their attempt to reach the Premier League.
First-choice goalkeeper Eldin Jakupovic had a nightmare against Sheffield Wednesday on Saturday, when he dropped a routine corner into his own net and then ended up in hospital with a facial injury.
Jakupovic, who was injured in a clash with Wednesday defender Reda Johnson, was stretchered off in a neck brace and taken to Hull Royal Infirmary.
Although the 28-year-old was later given a clean bill of health and is expected to play in tonight’s FA Cup tie at Leyton Orient, Bruce is anxious to address the goalkeeping situation once and for all.
“As a club we’ve had too many goalkeepers over the last couple of years,” said Bruce, with Hull having used nine different keepers in their last 80 Championship games. “I’ve said before it’s not a healthy situation and we’ll look to address it if we can.”
Mark Oxley became that ninth goalkeeper when he replaced Jakupovic in Saturday’s fixture.
The former Rotherham man is cup-tied, leaving only Jakupovic and youth team player Joe Cracknell available for the third-round replay at Brisbane Road.
Bruce said Jakupovic would be given a chance to redeem himself.
“He’s made a mistake for the (Sheffield Wednesday) second goal – he’ll know that,” said Bruce.
“But he’s a goalkeeper and sometimes they make mistakes; he’s been excellent over the past five or six weeks.
“The one thing you want to do when you’ve made a mistake is to go out and put it right at the first opportunity. Hopefully, he’ll be ready.”
Hull will be looking to bounce back from their shock 3-1 defeat in the Yorkshire derby, which dented their promotion hopes and saw Wednesday climb out of the relegation zone.
The result ended the Tigers’ eight-match unbeaten run and left them in the second and final promotion place on 49 points from 27 games – just two points ahead of Leicester, Crystal Palace and Middlesbrough.
Leyton Orient go into tonight’s match on the back of a 1-1 home draw with Crewe, which left them 15th in League One.
They were seconds away from beating Hull 1-0 at the KC Stadium earlier this month, but were taken to a replay when Nick Proschwitz equalised deep into stoppage time.
Last six games: Leyton Orient LWLDWD, Hull City WDWDDL.
Last time: Leyton Orient 1 Hull City 1, Division Three.
Referee: G Scott (Oxfordshire).