The 22-year-old was expected to become Atletico Madrid's first-choice number one following David de Gea's departure for Manchester United at the start of last season.
Yet he managed just three games last season before he was shunted back to the bench and, despite impressing during a subsequent loan spell at Rayo Vallecano, has been restricted again this term.
Needing back-up for Ali Al Habsi beyond the 40-year-old Mike Pollitt, Martinez has opted to get Joel in on trial, with reports in Spain suggesting he completed his first training session on Wednesday.
The youngster will not be allowed to feature in competitive games, although if he impresses a formal arrangement could be reached.
Any deal would bring to three the number of new faces brought in so far this month, with Honduras midfielder Roger Espinoza and Manchester United's on-loan striker Angelo Henriquez pencilled in to make their debuts in Saturday's FA Cup third-round tie with Bournemouth.
And, according to midfielder David Jones, it is not likely to be the end of Martinez's strengthening.
"The manager could well look to bring in more new faces by the end of the month, which he will obviously do with a view to making us a stronger group of players," said Jones. "Competition will be increased and places will be harder to secure. But modern-day football is like that and you have to be ready for it.
"If they can help the team and help us get the results we need, that is the main thing because we are not far away from getting the results we deserve.
"We all want to play every game but you don't want to get too comfortable in your position. You want the extra edge and the feeling you deserve to be where you are. The new guys will have to prove themselves so I am sure it will be an interesting and exciting few months."
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