Chelsea became the first top-flight club to announce fans would need proof of being fully vaccinated or having a negative test result in order to attend games this coming season.
Tottenham will trial the same measures for their friendly at home to Arsenal on 8 August.
Arsenal are likely to follow suit.
The Gunners face Chelsea in a friendly at Emirates Stadium on Sunday.
Clubs will be allowed to run trials in pre-season friendlies or Premier League games with the season beginning on 13-15 August.
But there could be a tightening of rules by 1 October.
Last week, vaccines minister Nadhim Zahawi said a Covid passport from then would mean fans “needing to show that they are fully vaccinated and proof of a negative test will not be sufficient”.
Chelsea say it will check fans’ covid status at the “point of entry” to Stamford Bridge and are confident that procedures will be in place to account for large groups of fans arriving shortly before kick-off.
But there is an acceptance from some clubs that it might not be possible to check all fans tickets once stadiums are at full capacity.
Football Supporters Association chairman Malcolm Clarke said it could lead to “chaos” at some clubs and told BBC Sport that fans with season tickets who are unable or unwilling to attend matches because of vaccine passports should be “entitled” to a refund.