The 24-year-old spent four months of last season on loan under Malky Mackay at Watford, with the Scot trying to make the deal permanent.
“I’m pretty sure there will have been conversations between Andrew and Cardiff,” said Boro boss Tony Mowbray.
“I know Malky very, very well and I’m pretty sure as soon as Malky got that [Cardiff] job he was on the phone [to Taylor].
“The boy’s got some decisions to make – that’s quite plain. And I think Cardiff are a different financial football club to Watford. They could probably match if not better what we could probably offer.”
And Mowbray also admires the Hartlepool-born player, saying: “Andrew, for me, is more than just a footballer – he’s three footballers, if you can say that.
“He showed that at the end of the season playing at left-back and in the games that the team won – they won the last four games – with Andrew at left-back.
“He spent most of the time in midfield, in a three-man, a four-man midfield, [so he’s] a very versatile player and I would like to re-sign Andrew Taylor, who is out of contract.
“But in any deal there’s more than just the manager and the player involved. There’s a few other issues that need resolving and we need to work our way through them.”