The New Zealander was on Wednesday confirmed as Lions coach for the third successive time after he led the tourists to Australia and New Zealand in 2013 and 2017 respectively.
The 55-year-old is standing down as Wales coach after 12 years once their Rugby World Cup campaign concludes in Japan later this year.
England's Rugby Football Union (RFU) have reportedly been interested in Gatland succeeding Jones but the Kiwi coach was having none of it.
"I can promise you 100 per cent I won't be coaching England," he told the BBC. "My understanding is Eddie is going to re-sign isn't he for a bit longer.
"My future is going to be the 2019 World Cup [with Wales], look at a few things in between, and then the start of my [Lions] role in August 2020.
"I'll focus 100 per cent on the Lions for those 12 months and then hopefully have an opportunity to go back to New Zealand and pick up something and then take it from there."
Jones is contracted to the RFU until 2021 but he could depart this year should England perform poorly at the World Cup in Japan.
Gatland has previously declared his intentions of returning to coach in New Zealand after spending more than two decades in coaching jobs in the northern hemisphere.
He said, after coaching the Lions, that he plans to come home after touring South Africa with the Lions in 2021.
"I would love to be involved with Super Rugby and to challenge myself with that," he told the BBC.
"I want to go back. I have been head coach with Waikato and won a championship there and I want to challenge myself with Super Rugby."
The 2021 tour will be Gatland's fourth with rugby's famous touring team after first working as an assistant under Lions legend Sir Ian McGeechan in South Africa in 2009.
Gatland will make history in joining McGeechan, who coached and played for the Lions, as the only coaches to have taken charge of the Lions on three consecutive tours after the Scot did so in 1989, 1993 and 1997.
The former Waikato coach dashed back to New Zealand last month to support son Bryn after the Highlanders pivot was ruled out of rugby for six months after a horror foot injury.
A long-term homecoming looks more possible after the 2021 Lions tour, which will be Gatland's fourth with rugby's famous touring team after first working as an assistant under Lions legend Sir Ian McGeechan in South Africa in 2009.
The Lions lost the 2009 series 2-1 to the Springboks, so Gatland can target becoming their first coach to go unbeaten in series with the southern hemisphere's big three, South Africa, Australia and New Zealand, having beaten the Wallabies 2-1 in 2013, then drawn 1-1 with the All Blacks in an enthralling 2017 tour