Jenkinson, 23, has spent the last year on loan at West Ham having slipped down the pecking order at Arsenal, with Arsene Wenger heavily relying on the likes of Mathieu Debuchy, Hector Bellerin and Calum Chambers at right back this season.
The England Under-21 international has thrived at Upton Park, playing on a regular basis for the Hammers and producing excellent displays under previous boss Sam Allardyce.
Now though, Jenkinson has returned to Arsenal given that his loan spell has now ended, with his future up in the air given that there are plenty of other right backs in the mix at the Emirates Stadium.
The Telegraph now report that Sunderland boss Dick Advocaat is pushing to sign Jenkinson on a season-long loan deal for next term, with a £2m loan bid - with no obligation to purchase him next summer - is now on the table.
That could be something of a blow for new West Ham manager Slaven Bilic, who will be looking to shape his squad quickly ahead of the Europa League preliminary rounds this summer.
The Mirror suggest that Bilic will turn to his compatriot and former Tottenham defender Vedran Corluka, with the Croatian seen as an alternative to Jenkinson - and given that Bilic signed Corluka for Lokomotiv Moscow and managed him as Croatia's national team boss, Corluka may well find his way to Upton Park instead of Jenkinson this summer.