"From where we sit at the moment, based on the information we have, based on the fact that every day there are less and less cases in the UK everyone is getting increasingly confident that the tour will happen at some point," ESPNCricinfo quoted Grave as saying.
"Our board meeting is on May 28. If the tour has to take place, by the start of June we would need to have the CWI board approval and support by then in order to get the charter flight logistics in place and select the players," he added.
Grave has also suggested that the Test series could take place in Southampton and Old Trafford in a bio-secure environment.
He also made it clear that none of the Windies players have yet told the board that they do not feel like touring the UK.
"The current plan we are working to has the three Tests starting on July 8, July 16 and July 24. As for the venues, we believe it could be the Ageas Bowl and Emirates Old Trafford, but we haven't had that formally confirmed yet," Grave said.
"Players have concerns, but certainly I have had no feedback that a player has said, based on what we know now and what we have shared, that they won't tour. No-one has directly come out and said they won't tour, but we haven't formally asked them yet," he added.
Last week, the England and Wales Cricket Board (ECB) allowed its players to resume training as the country plans for a return of international cricket.
With this, England became the first nation to commence training for its players after being hit by the coronavirus.
The board said by utilising venues across the country for individual sessions, it will be able to provide a controlled environment that ensures adherence to safety protocols and social distancing measures for players and staff as set out by the government.
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