France international Gomis, who became frustrated at a lack of game time prior to Wilfried Bony's departure to Manchester City, sparked intense speculation last month when he spoke of the club's alleged broken promises on French television.
Crystal Palace and Schalke were among the clubs linked with a move for Gomis, but the 29-year-old took to Twitter on Monday to reassure fans of his happiness in south Wales and declare himself a 'Jack' - a name used to describe people from Swansea.
And Jenkins believes Gomis has the ability to move on from the uncertainty surrounding his future.
"I am sure he will settle here," he is quoted as saying by the South Wales Evening Post. "He is happy living here and so are his family, and he takes on board the fact that we made a big effort and commitment to bring him here last summer.
"Yes, ideally he would like to have played more in the first half of the season, but now he has the chance to do that and I expect him to show what a good player he is.
"When we signed Gomis, we felt he would be the right type for us.
"We didn't know how long Wilfried Bony would be with us and we took the decision to bring Gomis in.
"There were one or two bits of interest in him during the window, but you can understand that because there are not many strikers out there. He will always stand out as a player people will look at.
"But now he has the chance to start regularly and to show everybody what he can do."
