The 24-year-old has been without a club since Widnes Vikings went into administration last month.
He joined Widnes from PNG Hunters ahead of the 2018 season and made 12 appearances in his debut campaign.
Albert has five Test caps and played for PNG in the 2013 and 2017 World Cups, scoring a try against New Zealand at Headingley in the tournament six years ago.
He began his career with Lae Tigers and in 2014 became the first Papua New Guinean player to be signed by an NRL club without having played junior rugby league in Australia when he signed a deal with Penrith Panthers.
He played for them at National Youth Cup level and in the Intrust Super Premiership, before joining PNG Hunters in 2017.
Rhinos’ director of rugby Kevin Sinfield said: “We are happy to give Wellington an opportunity to train with us.
“He has found himself without a club through no fault of his own and we are without a number of first choice forwards at present.
“There is no formal commitment from either party at this stage.
“Wellington would be free to join another club at any point.”
Albert had been set to join Toronto Wolfpack but that move appears to have fallen through.
If signed, he would be exempt from Rhinos’ overseas quota.