But Johnson has assured supporters he has no intention of moving on again.
He told the Bristol Post: “Why would I want to leave when I’m so happy?”
Johnson added: “I want people to know I’m doing nothing to put myself in the shop window or attract interest from outside.”
Bristol City are one of four teams to have beaten Di Matteo’s Villa this season, three of whom – including Luton’s Nathan Jones and Preston’s Villa old boy Simon Grayson – have had their manager’s name linked with the job via various media reports.
Johnson, 35, who has been in charge at Ashton Gate since February, is already into his third job in football management, having previously been at Oldham – from March 2013 to February 2015 – and then Barnsley for just under a year.
Analysis – BBC Sport’s Pat Murphy
“Aston Villa have interviewed no-one yet for the vacant manager’s job, with none planned until next week.
“Background checks and due diligence are still being done on the favoured candidates and Villa have to consider if an approach should be made to clubs employing any of those.
“Money is no object in terms of compensation but Villa realise an approach may be knocked back.
“I understand Burnley’s Sean Dyche and Huddersfield’s David Wagner attract Villa, while free agent Steve Bruce remains the bookies’ favourite.
“Eventually a shortlist of three will be drawn up and the owner Dr Tony Xia will take the final decision after consultations with the recruiting panel – chief executive Keith Wyness, technical director Steve Round and former Villa manager and player Brian Little.
“Caretaker Steve Clarke is also being considered and Villa are relaxed about the possibility of him being in charge for the next league match, at home to Wolves a week on Saturday.”