While Jenkinson has signed a one-year deal at Upton Park, there are unconfirmed reports he has secured a new five-year contract at Arsenal.
West Ham co-chairman David Gold revealed on July 2 the club were “working hard” to sign the Gunners defender.
But on Monday night, it was reported the Hammers had got their man, albeit on a season-long contract.
Jenkinson found his first-team chances at Arsenal blocked by Calum Chambers and Hector Bellerin and spent last season on loan at the Hammers.
The 23-year-old made 29 Premier League starts for them.
Gold had previously stated the club were keen to bring in players to play the “West Ham way” and last week stated: “We are very working hard to try and get Carl back here.
“We recognise this is an important season for us, being in Europe, and we will go the extra mile to ensure we have got the strongest possible squad we can muster.”
Mathieu Debuchy and Bellerin will be fighting for the right-back position in Arsene Wenger’s team and another impressive campaign could see Jenkinson return to the Emirates.