According to journalist Ángel García, the 21-year-old midfielder's next move could take him back to Spain, a country he already knows from a loan spell at UD Las Palmas.
Bajcetic made 14 appearances and scored one goal during that loan.
Rayo Vallecano, Sevilla and Elche have all shown interest in signing the midfielder, according to the report.
Bajcetic was born in Vigo and came through the youth ranks at Celta Vigo before joining Liverpool.
He has won five caps for Spain's Under-21 side.
The midfielder has made 22 official appearances for Liverpool's first team, scoring one goal.
His lack of involvement last season has left him surplus to requirements at Anfield as the club shapes its squad for the new campaign.
A move to LaLiga would represent a return to Spanish football for a player who has already shown he can perform at that level during his time with Las Palmas.
Rayo Vallecano, Sevilla and Elche will now have to compete to convince both Liverpool and the player himself that their project offers the best next step in his career.