The Austria international was a cult hero at Athletic between 2006 and 2010 before he joined West Brom and then Bundesliga side Hamburg.
Martinez has lost centre-back Ivan Ramis to injury for the season and wants to bring in Scharner as a replacement.
"Over the next 48 hours I am hoping to bring one or two players in and Paul is one of them," Martinez said.
"We had the injury to Ivan Ramis, his season is over and we are very short at the back. Antolin Alcaraz is not back yet, so I am trying to bring some cover in at centre-half.
"Paul could be a perfect replacement because he knows the football club and the Premier League.
"He is someone who could fit in at centre-half and with the way we play.
"The fans would welcome him with open arms and it is something that I am hoping to have positive news on over the next 48 hours."