The players in question are believed to include Sam Burgess and Danny Care.
Reports suggest they invested in an oil-drilling firm and lost a combined total in excess of £100,000.
An RFU spokesperson said: "The RFU takes the allegations extremely seriously. It is an internal matter and we are taking the appropriate action."
England boss Stuart Lancaster and his coaching team are understood to have had no knowledge of Tennison's alleged shares advice.
Lancaster's side endured a poor World Cup on home soil as they were eliminated at the group stages.
Their performance - and Lancaster's position - are being reviewed by the RFU and chief executive Ian Ritchie will report to a management board meeting on 17 November.
Burgess has since returned to rugby league, having left Bath to move back to Australia to join South Sydney Rabbitohs.
England kit man Dave Tennison is under investigation by the Rugby Football Union over claims he urged players to buy shares which fell in value.