The FA have repeatedly said they will name a replacement for departed coach Fabio Capello, who resigned in February, towards "the back end of the season" and have appointed Under-21s coach Stuart Pearce as the caretaker.
Other issues, such as who will captain the team, are also unresolved and both Neville and Venables said at the Soccerex European Forum that time was now running out with the tournament starting on June 8.
Neville, the former Manchester United defender, who played 85 times for England and is now a respected TV analyst told delegates at the convention that the uncertainty surrounding the England team was worrying.
"There's something brewing at the moment. There are situations still not being dealt with. There are player situations going into the tournament, there are manager situations... and it's going to come pretty quickly. It's April next week," Neville said.
"There's manager, there's player, there's captain, there's (John Terry) court case... it's not going to go away.
"And the FA at this moment are in a difficult position as always. If they actively go in there now and do something then it will be like a bull in china shop.
"But I do feel the vultures are circling. The England manager has started being mentioned again this week, we are going into a tournament - and who is taking the team, what tactics are we going to play, what principles?
"The end of the season is four weeks away. They say they are going to wait until the end of the season - that's May 13, and I think they should make their decision as soon as possible."
Venables, 69, who managed England from 1994 until 1996 and guided the team into the semi-finals of Euro 96, told delegates that he believed that Tottenham Hotspur manager Harry Redknapp was the right man for the job.
"I think Harry is the candidate, he has done such a capable job at Tottenham," Venables said.
"But it's gone on too long for me and no one seems to really know what's going on apart from those that are in the seat. They may know what's going on, I don't know whether they do."
And he agreed with Neville that a decision should be made as soon as possible.
"The more time you can get in that job the better. There's nothing for you doing it off cold. You need everything in your favour because it's such a big job. They have to get someone in there straight away do to something about it.
"I don't see any benefit at all in coming in at the last moment with a rah-rah-rah.
"I think Harry is capable of saying "I'll go". He has got Tottenham into a very, very strong position, they had a bit of a blip but they seem to be over that and he qualifies as England boss by ticking a lot of the right boxes."
Venables also said he was strongly in favour of an English manager succeeding Capello.
"I can't get to grips with that (foreign England managers). I never have. You can go for anyone in the world for the Champions League and Premier League.
"But internationally it's like what went on in wars – you can't trade generals you have to go for your own. Spain and Germany always go for their own. I would rather lose with our own people. Some of our guys deserve a chance and can do it."