The RFL are working with DCMS and Wembley Stadium to try and ensure as many supporters can attend with Castleford Tigers and Featherstone Rovers both set to be in action that day.
The Tigers take on St Helens in the Challenge Cup final, hoping to win their first major trophy under Daryl Powell in what will be his final season as head coach at the club.
Rovers will be back at Wembley for the first time in nearly four decades, with York City Knights standing between them and the 1895 Cup.
Mark Foster, the RFL’s Chief Commercial Officer, said: “We have had talks with the Government in recent weeks and particularly in the last week since the semi-finals, and these continued over the weekend and today [Monday].
"With further discussions planned during the next couple of days, we hope to make a further announcement on ticketing arrangements before the end of the week.
“We have deliberately held back a proportion of tickets at Wembley Stadium, even for a limited capacity, to ensure that some supporters of the four clubs involved will be able to attend.
"With over four weeks to the finals the timescale is tight, but we still have enough time to ensure fans are able to purchase tickets, deal with any requirements to attend, plan their journeys and have a safe and enjoyable experience on match day.”
Wimbledon tennis finals, which will be staged on July 10 and July 11, are set to be the first outdoor sporting events with capacity crowds despite the delaying of lockdown easing.
Four Euro 2020 matches at Wembley, including the July 11 final, will be able to operate with 50 per cent capacity.
Decisions are pending on the Challenge Cup final, The Open and the British Grand Prix. With the Challenge Cup and 1895 Cup finals set to be played on July 17 and the full lifting of restrictions now set to for July 19, it is unlikely Wembley will see a capacity crowd as St Helens, Castleford, Featherstone and York fans make their way to the capital.