Two statements on Wednesday afternoon followed similar announcements over the past week from Sussex and then Somerset.
ECB chairman Colin Graves identified a "watershed moment" last month when he triggered a postal ballot to endorse the amendment to the national governing body's articles of association.
The vote, involving all 18 first-class counties, the MCC, Minor Counties Cricket Association and 21 recreational boards, must be concluded within 28 days.
But since the ECB's executive board resolved to dispatch the paperwork on March 28, indications are already highly favourable that the counties will give the go-ahead - on a one-off basis - for a competition consisting of just eight teams, rather than all England's affiliated professional clubs, to begin in 2020.
Yorkshire's press release read: "The board of the Yorkshire County Cricket Club has unanimously approved a change to the ECB articles of association in order to facilitate a new T20 competition in England and Wales, which will commence in the year 2020.
"The club has attended many consultative meetings with the ECB over the last 18 months, and Gordon Hollins, the chief operating officer of the ECB, was present at a members forum in the Long Room on Friday April 7, when members were able to discuss the implications of the new competition."
Headingley is expected to be among the venues most likely to host a team in 2020, but Yorkshire chairman Steve Denison is taking nothing for granted yet.
He said: "This is not just about a new city-based T20 competition.
"It is all to do with the ECB's 'Cricket Unleashed' strategy, which is setting out to broaden the appeal of cricket to a much wider audience.
"Cricket needs to attract boys and girls to the game through the All Stars Cricket programme, and through greater exposure on mobile devices, tablets, as well as television.
"Headingley hopes that it will be one of the chosen grounds for the new competition, but the club understands that this may well be predicated on the development of the new North/South Stand, in conjunction with Leeds Rugby."
Two days ago, Somerset's statement included an understanding from their club chairman, Andy Nash, that his county "will be provided with the opportunity to stage matches in the new tournament".
Each club will receive a £1.3million annual share of the new competition's revenue and the ECB needs a vote of at least 31 in favour from the 41 member organisations consulted.
Nash added: "It would be remiss not to record the considerable unease and concern of our general committee in regard to some potential longer-term implications of this decision.
"There are clear risks but ones that on balance we consider worth taking.
"In short, Somerset wants to see 18 thriving counties, not an elite group of eight with 10 stragglers trailing in their wake."
In Leicestershire's statement, club chairman Paul Haywood signed off by making the same point.
It read: "The retention of 18 first-class counties is sacrosanct, and the ECB have agreed this."
