The International FA Board has recommended the trials, which would allow referees to seek assistance from an extra official on decisions relating to goals, red cards, penalties and cases of mistaken identity.
Chelsea interim manager Guus Hiddink and Manchester City boss Manuel Pellegrini are among those to pronounce themselves broadly in favour of the plan.
But bosses also warned the continued introduction of technology could have a negative impact on the natural flow and excitement of the game.
Hiddink said: "Any kind of help is welcome in the modern age and football is such a conservative game, but it must not be overdone.
"You can have some tests, especially when it's in key situations - when there's a goal coming which is not allowed
[But] it must be in key situations."
Pellegrini appeared to go further, suggesting he would like to see managers given the chance to challenge refereeing decisions in the same way as tennis or cricket.
The Chilean said: "I think each manager can have, similar to tennis, one play in each 45 minutes to review - just one - and about important things, about red cards, about penalty, about goal
"I think it is very important - the same as goal technology - not to use the technology to change the criterion of the referee, but just if it is a mistake."
Leicester boss Claudio Ranieri also backed the proposal, saying: "I would want it, yes - in the right way
You can check immediately, in 10 seconds you can say
The opportunity would help the referee, help everybody.''
Scottish FA chief executive Stewart Regan suggested this morning that the proposal could also be trialled in next season's Scottish Cup.
And the news was welcomed by Rangers boss Mark Warburton, who said: "Yes I am in favour of video technology
"I think the pressure on match officials is getting ever more intense and I think every help we can give them, we should.
"I understand the concerns about losing the rhythm of the game but with technology now there is no delay
However, some leading bosses reacted with more caution, with Roberto Martinez admitting he is "not a big supporter" of video technology.
The Everton boss said: " Goal-line technology has been an incredible help to referees but I think we should allow referees to manage and understand the game
"If they can't see something and they need help then video technology could be helpful but I wouldn't like to take human error away from football as it is part of the game."
And Tottenham boss Mauricio Pochettino echoed Martinez's sentiments, warning the introduction of technology risked interrupting the flow of the game
Pochettino said: "The good things in football are that we can speak about it after game; whether it was a red card, a yellow card, or it was offside
"Maybe football needs to keep some things important, the human decision need to exist in football.
"Okay, we know it is more professional these days
Football needs to keep a sense of the past."
Source : PA