The Spain international suffered the injury in a cycling accident during a family holiday in Italy last month and is likely to miss the start of the new Premier League season as a result.
Clement remains hopeful that Llorente will be ready in time for their season opener against Southampton on August 12, but admitted that he is unlikely to be fully match fit.
"This is a period where we have to be very careful that Fernando is not moving the arm. It's difficult to know at this stage exactly when he will be fit to play. There is a chance that he could make the first game of the season, but I don't think he will be in the kind of shape the rest of the squad will be in because they will have done six weeks [of training] and he potentially will only have done two," he told the club's official website.
"We have to see. It's not impossible that he could be ready for the start of the season. He had another X-ray on Monday which showed the bone is healing well. We need him to keep it in that position for a further week and then he will be X-rayed again.
"If it's okay, he can begin to move it a little bit more at that point but there is no point him coming to America. There is work he can do, but it's better for him to use the facilities we have got at Fairwood. He can do one-to-one work there while we are away."
Swansea will travel to America on Thursday ahead of a three-match tour which will see them take on Philadelphia Union, Richmond Kickers and North Carolina.