The 29-year-old joined Leyton Orient on a free transfer in the winter of 2020 and has been a key player since his arrival at the London club.
Vigouroux featured in 45 league games for Leyton Orient last season and kept 24 clean sheets to help the League Two outfit earn promotion to League One.
Vincent Kompany is determined to build a squad that will be able to stay in the Premier League next season.
Burnley are looking to strengthen their goalkeeping department after the departure of Will Norris.
And they want to add someone with experience to the squad to provide competition for Arijanet Muric.
Vigouroux’s contract with the freshly promoted League One outfit ends at the end of June and the player is not planning to not renew his contract.
It has been claimed that Burnley are set to land the goalkeeper on a free transfer in the summer.
And it remains to be seen whether Vigouroux will be able to steal the the number one spot from Muric when given an opportunity.