The 20-year-old was part of the England squad to win the U20 World Cup in South Korea in June, but finds himself on the fringes on Merseyside.
Having made just two appearances for the senior side last season, hampered by a troubling back injury, Ojo is in need of regular first-team football in 2017/18.
Klopp is believed to enjoy positive relationships with both Rafa Benitez and Garry Monk, which should allay his fears that Ojo would stagnate further if he left Anfield.
Monk is currently lacking in genuine wide options as he looks to inspire Boro to promotion at the first time of asking.
Ex-Barcelona youth Adama Traore is attracting interest from a host of clubs both in England and abroad, with the Magpies among those touted as an option for the 21-year-old.
If Adama did make the move to St. James’ Park, Ojo could expect to play a significant role in Boro’s push for a return to the Premier League.
Liverpool have seen many of their youngsters shine on loan in the Championship in recent seasons, including Danny Ward at Huddersfield Town in 2016/17.