United, battling to get back into the Premier League's top four after finishing seventh in 2013-14, were looking to part ways with Cleverley, eventually shipping the midfielder to Aston Villa on loan on deadline day.
Martinez, who linked up with the England international when he was loaned from United to Wigan for the 2010-11 season, said he was a big fan of the 25-year-old.
But bringing him to Goodison Park was made tougher by United's seeming unwillingness to bolster a rival to their UEFA Champions League ambitions, according to Martinez.
"I am a big admirer of Tom," the Spaniard said.
"That was an opportunity that we explored on the last day of the window but it couldn't happen for different reasons. I don't think there was just one aspect.
"It's the parent club that makes that decision.
"Maybe they just didn't want to send a player on loan to a club that is going to be challenging for things and finished above them last season, I don't know."
Martinez disagreed with the notion there was bad blood with United, after former Everton boss David Moyes' stint at Old Trafford saw him successfully poach Marouane Fellaini and try to also bring across Leighton Baines.
"I don't think there is any bad feeling between the clubs," said Martinez.
"The chairman [Bill Kenwright] has a very good relationship with Manchester United and, as a football club, we are not carrying any sour grapes about anything.
"Our information was that the player [Cleverley] wasn't available on loan full stop. You can look at the table from last season and we finished fifth and they finished a bit lower, so from that respect you can understand.
"But, at the same time, Aston Villa have started the season really well. They are going to be a strong team.
"But we can't comment on that, it is down to Manchester United. A situation with Cleverley came up that I was happy to explore. In the end it didn't work out."
