Tierney is a highly sought after defender following his impressive displays in a Celtic shirt across the past few seasons.
However, their first two bids for the Scotland global - both thought to be less than £19m - were rejected by Celtic.
Jenkinson has 18 months left on his Arsenal deal and Celtic are looking to increase competition in defence after losing Mikael Lustig and Cristian Gamboa.
At such a young age, Tierney's experience of working within a winning culture, alongside his nine appearances in the Champions League, makes him a player that would bolster Unai Emery's defensive options significantly. Tierney is a solid defender, who likes to bring his team forward with his dribbling ability, and the hope at Arsenal would be that he can make the same sort of impact in the Premier League as another Scottish left-back - Liverpool's Andrew Robertson.
The Scottish global has played 170 games for Celtic, despite his young age, and has even made his national team debut for Scotland, playing 12 games for his country.
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Sport states it hardly took any time for Ernesto Valverde's side to agree a deal of €24m net per season. Should they turn down the opportunity to land him, then he will consider the option of Real Madrid.
Reports suggest that Arsenal are prepared to offer a lucrative long-term deal with wages of £70,000-a-week, representing a significant upgrade on Tierney's deal at Parkhead.
"I don't think that will be even close to tempting Celtic into selling. It's one of the top sides in England coming and saying we're going to take your best player for a pittance".
On Monday several publications have claimed that Arsenal have made a second offer for Kieran Tierney and this time they are willing to shell out £20 million for the left-back.
Apart from Arsenal the only other club which have shown a serious interest in signing him are Leicester City which is now being managed by former Bhoys boss Brendan Rodgers.