Mauricio Pochettino is a keen admirer of Rodriguez, whom he worked with during his time at Southampton and the Argentine manager is hoping to work with him at Tottenham.
A move for the 25-year-old has been on the cards for much of the summer despite the fact that Rodriguez is ruled out until at least October through injury.
The Saints are interested in acquiring Tottenham's Andros Townsend and the two clubs could come to an agreement which would see both players swap clubs before tonight's transfer deadline.
Townsend has struggled to impress Pochettino and his mistake which allowed Alberto Moreno to score Liverpool's third in yesterday's 3-0 loss to Liverpool will not have helped his cause.
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Rodriguez played eight more games than Townsend last season, and although the Saints man can play as a striker, he was mainly deployed out wide.
He scored 15 goals last season with three assists, while Townsend registered just one goal without a single assist.
The Spurs winger has struggled to rediscover the form he showed 12 months ago which earned him his first England call-up and he is now not as much of a threat as Rodriguez.
The Southampton star's versatility in the final third would make him an excellent addition if Spurs can complete a deal, even if Townsend was to leave.
The Lilywhites are also linked with a move for Manchester United's Danny Welbeck, and it remains to be seen if Rodriguez will be approached before the deadline tonight.