The former Warrington Wolves player, 27, joins with immediate effect and will fly out with the squad to Spain next week.
He will spend the full 2020 campaign at Emerald Headingley and look to re-establish himself as a Super League player.
Evans played more than 100 games for Warrington during seven seasons in the top-flight including the 2016 Challenge Cup final and Grand Final.
He dropped down to the Championship with Leigh Centurions in 2018 but, after they hit financial problems, moved on to Odsal last term.
Rhinos head coach Richard Agar had said before Christmas he would like to bring in one or two more new players before the season kicks off against Hull FC on February 2.
“We are pleased to have brought a player of Rhys’ undoubted experience to the Rhinos to bolster our squad,” said Leeds director of rugby Kevin Sinfield.
“He can cover a number of positions in the outside backs.
“Last season we were able to bring James Donaldson during pre-season to increase our quality in depth and we hope that Rhys will have a similar impact in 2020.
“He will immediately join up with the group and will join the squad for the pre-season camp in Spain next week.”
Evans scored three tries in 15 appearances for Bradford last term and Bulls boss John Kear revealed: “Rhys had a release clause in his contract so he was entitled to move to a Super League club.
“By agreeing the loan proposition with Leeds he now has an opportunity to establish himself.
“As part of the agreement the Bulls will also get a couple of Leeds players, which will be announced later in the month. Rhys is a really smashing lad, with a strong Super League pedigree.
“On a personal level I am pleased for him that he has this opportunity.
“The overall arrangement represents sound business for our club.”
Rhinos face Bradford at Headingley on Sunday January 12 in Jamie Jones-Buchanan’s testimonial game which is also now a fundraiser for Rob Burrow’s fund, following the half-back’s diagnosis with motor neurone disease.
Meanwhile, Castleford Tigers are mourning the loss of one of their Hall of Famers, Johnny Ward.
The forward played more than 250 games for the club and was also part of the Great Britain squad that won the Ashes in Australia in 1970.
He helped Castleford win the Challenge Cup in 1969 and also scored twice when they famously beat the touring Kangaroos six years earlier.