The 21-year-old looks set to join on-loan Chelsea forward David Datro Fofana as a January arrival at Turf Moor.
Footage on social media appeared to show Esteve in attendance at the 3-1 loss at Manchester City on Wednesday.
And when asked at a press conference ahead of Saturday's home clash with Fulham how close the Frenchman was to joining, Clarets boss Kompany said: 'If he's in Manchester, then pretty close.
'Walking up here, there's still nothing official, so until then... but he's in the building and hopefully we can announce it soon.
Vincent Kompany has stated that Maxime Esteve is 'in the building' as Burnley close in on a deal to sign the Montpellier defender
Kompany added he was 'hopeful' that Burnely could 'announce the deal soon'
The 21-year-old has made 14 appearances for Montpellier this season across all competitions, returning one goal and an assist for the Ligue 1 outfit
'I hope if it does happen that everybody will see what we saw in him, but until the ink is on the paper, I'm not going to say too much.'
Esteve has made 14 appearances across all competitions for Montpellier this season, returning one goal and an assist.
He has also been capped at international level for France, having made one appearance for the French U21 side at the end of last season.
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