Robins is keen to make a swift return to the dugout after being sacked by Coventry City in November.
He also attracted attention from West Bromwich Albion, who lost previous boss Carlos Corberan to Valenica.
However, Robins favours Stoke, who sacked Narcis Pelach on Friday, December 27 after just 19 games in charge.
Pelach, who had previously coached at Huddersfield Town, managed just three wins in that time and lost nine.
They did, however, win their first match without him on Sunday as club legend Ryan Shawcross, currently their Under-21s' head coach, took interim charge.
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The Potters, who spent 10 seasons in the Premier League between 2007 and 2018, currently sit 18th in the EFL Championship.
That keeps them just outside the relegation zone, but a run of form could easily propel them into the middle of the pack.
Robins was a popular figure at last club Coventry City, despite being replaced by Frank Lampard.
In 2022/23 he led the Sky Blues to a fifth place finish and only lost out to a place in the Premier League on penalties when they were beaten by Luton Town.
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Meanwhile, last season they finished ninth and caused a stir when they came within a whisker of beating Manchester United in the FA Cup semi-final at Wembley.
They came from 3-0 down to level the scores but once again lost on penalties.
However, despite being fondly thought of, a run of form which had them in midtable was enough to seem him sacked.
The 55-year-old is a renowned Football League manager with spells at Rotherham, Barnsley, Huddersfield, and Scunthorpe United.
Coventry is where he excelled, though, leading them from League Two to the cusp of the Premier League.
Meanwhile, he also won the EFL Trophy with the club in 2017.
As a player, Robins represented Manchester United, Norwich City, and Leicester City, amongst others.
With the Red Devils, his famous 1990 FA Cup third round goal against Nottingham Forest is often cited as a sliding doors moment for Sir Alex Ferguson.
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The legendary manager was under-pressure but the strike put them into the fourth round of a competition they would eventually win.
Man United had been 15th in the First Division and already knocked out of the League Cup.
However, the FA Cup triumph then sparked a turn in fortunes which saw them win 13 Premier League titles, seven FA Cups, and two Champions Leagues.