The former Barcelona legend jetted into Edinburgh on Sunday night to investigate a possible move to the Scottish Premier League club.
Amor joined the rest of the team as they returned to training on Monday morning and the former Spain international will spend the next 14 days at the club's City Stadium.
The former Fiorentina and Villarreal star has won five league championships and a European Cup during an illustrious career and the SPL could be his next stop at the age of 35.
"
Considering what he has achieved, he'll be the biggest name ever to come to Scottish football
"
Livingston manager Jim Leishman
Leishman will assess his fitness before beginning negotiations next week in the hope of signing the player at least until the end of the season.
"He will be training with us for a fortnight and we will speak to him next week about joining the club," he said.
"He has a great track record and he is a fantastic player and he seems like a really nice guy.
"He is out training with everyone at the moment, so we will see what his fitness level is like first and then speak to him.
"Considering what he has achieved, he'll be the biggest name ever to come to Scottish football."
Amor spent 10 years with Barcelona, followed by two at Fiorentina before his return to Spain with his last club, Villarreal.
"I haven't played for a while, so I want to get back into competition and see what happens," said the player who had a proposed move to the USA fall through.
Leishman will also hold talks with defender Michael Hart and goalkeeper Javier Sanchez Broto towards the end of the week in a bid to end the uncertainty over the future of the pair at Livi.
Hart is wanted by new Aberdeen boss Steve Paterson, who is keen to hand a return to Pittodrie to the defender who failed to hold down a first-team place during his first spell at the club.
Spanish goalkeeper Broto is out of contract at the end of the season and has already admitted he is keen to move on to a bigger club.
Newspaper speculation has linked Broto with SPL rivals Kilmarnock.