The midfielder has been a free agent since leaving Bournemouth, where he managed just 14 appearances during his second stint on the south coast.
Wilshere last played in the Premier League at West Ham , where he was paid £3.3m to cancel the last year of his contract after making just 19 appearances in two years for the Hammers.
Wilshere was once one of the best prospects in world football but a series of injuries derailed his career and limited him to just 34 appearances for England.
It was on national duty that the 29-year-old met Rooney, who skippered the Three Lions when Wilshere was selected for the 2014 World Cup and 2016 European Championships.
The pair became friends and Rooney, who has been in charge of the Rams for the last year, invited Wilshere to train with his side earlier this month.
It's unclear if Rooney would consider signing the former Arsenal ace but Wilshere is keen to impress as he searches for a new club.
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