The 30-year-old has been a semi-regular figure for Vincent Kompany's side in the 2023-24 term, making 21 top-flight appearances and providing two goal contributions.
Taylor joined the Lancashire club from Leeds United in the summer of 2017 and has since made 213 appearances across all competitions, finding the net just once.
As alluded to earlier, that solitary strike for the Clarets arrived this season, when the left-back scored the opener in a 2-1 loss at Bournemouth on October 28.
Speaking to Burnley Express during this month's international break, Taylor remained tight-lipped when questioned on his future at Turf Moor past the conclusion of the season, claiming that he has not given the situation much thought.
"No, not from my side. At the start of the season I wasn't playing, so I'm just happy to be playing now every week," said Taylor when the topic of his contract situation arose.
"I'm not really concentrating on anything past the summer, I'm just happy to be playing every week.
"I'm enjoying my football and trying to do the best for this club and get us as far up the table and out of trouble as much as I possibly can."
At the beginning of the current term, Taylor was not the first-choice option at left-back for Burnley, with natural right-back Vitinho often operating on the left flank.
However, the 30-year-old has been able to fight his way back into contention for Kompany, although the veteran is struggling to guide his side towards safety.
Even with the four-point deduction handed to Nottingham Forest earlier in the month, Burnley remain 19th in the Premier League standings, five points behind 17th-placed Luton Town.
With Taylor's contract expiring in the summer, the Yorkshire Post is reporting that there is expected to be plenty of potential suitors for the defender.
It is understood that one of those could be Championship side Leeds, who are on course for an immediate return to the top flight under the stewardship of Daniel Farke.
The Whites were said to be keen on a move for Taylor during the summer of 2023, however a move from Lancashire to Yorkshire never materialised for the player.
After navigating the first three months of 2024 unbeaten in the Championship, Leeds have moved to the top of the second-tier table, one point ahead of Kieran McKenna's Ipswich Town in third.