The Welsh return to action, after finishing second in the 2018 Six Nations, with a Test match against South Africa in Washington DC on June 2 before they travel to Argentina for a two-game series.
Gatland has decided to rest regular skipper Alun Wyn Jones for the tour and has instead opted for Cardiff flanker Jenkins and Newport lock Hill to be his on-field leaders, with the duo having proven themselves in the Pro 14.
Gatland has only named one uncapped player in his 31-man touring party, Cardiff Blues scrum-half Tomos Williams, and has left a few familiar faces behind as he looks to improve his squad depth ahead of next year's World Cup.
Taulupe Faletau, Leigh Halfpenny and Liam Williams miss out on selection, having been part of the British & Irish Lions tour last summer, but wing George North and No.8 Ross Moriaty, who also went to New Zealand with the Lions, are included.
Experienced Bath lock Luke Charteris is handed a recall as Gatland looks for players on the fringes of the squad to make a big statement as the countdown to the 2019 World Cup continues.
Gatland said: "This summer is a great opportunity for this squad to build on their experience, get Test match game-time under their belt and face Rugby Championship opposition.
"We have picked a squad of 31 to try and give players as much game-time as possible.
"It shows the depth we have built in recent years that we have just one uncapped player in the squad in Tomos, but he has been in a number of squads so knows the environment well.
"We have picked Cory and Ellis as co-captains. They are both leaders and have experience captaining at regional level so this is an opportunity for them to build on that."
Forwards: R Evans, W Jones, N Smith, E Dee, R Elias, T Francis, S Lee, D Lewis, A Beard, Bradley Davies, Seb Davies, L Charteris, C Hill (co-captain), J Davies, E Jenkins (co-captain), R Moriarty, J Navidi, A Shingler
Backs: Aled Davies, Gareth Davies, T Williams, G Anscombe, R Patchell, H Parkes, O Watkin, S Williams, J Adams, H Amos, S Evans, G North, T Prydie