Arsenal boss Unai Emery has made Tierney one of his top targets this summer.
Tierney is capable of playing on either flank and is seen, not only as a replacement for Nacho Monreal, but as cover for Hector Bellerin who will miss the start of the season through injury.
The Spaniard though is dealing with a relatively modest transfer kitty of £45m and must consider his options carefully.
The full-back would cost the Gunners around £25m, but Emery is prepared to include Jenkinson in an effort to get the deal over the line according to the Daily Express.
Newly appointed Celtic boss Neil Lennon is a fan of the Arsenal fringe defender and may be swayed by his inclusion in the deal.
Celtic’s academy graduate has previously been linked with Manchester United, but remains at Parkhead.
Arsenal though may be about to test the Scottish champion’s resolve with a player plus cash offer in the coming days.
Tierney was a huge success story of the Brendan Rodgers era and continues to be an integral part of the Celtic squad under Lennon.
The 22-year-old has already played his part in three straight treble triumphs in Glasgow.
Arsenal’s defence is in major need of reinforcements and was their Achilles’ heel in their failed pursuit of Champions League football.
Emery’s side shipped more than a goal a game – including 11 in their final five games of the season.
Saint Etienne’s William Saliba is another player the Gunners are tracking, but his £30m price tag is proving problematic.
The French club are also reluctant to let him go and have proposed a deal that sees him sign, but remain at the Stade Geoffroy-Guichard on loan for another year.
Emery though is not keen on spending such a huge chunk of his transfer budget on players he will not reap the rewards of for another 12 months.