The company already back Scottish teams Edinburgh and Glasgow through shirt sponsorship and they are now said to be targeting Welsh sides Cardiff Blues, Ospreys, Scarlets and Newport Gwent Dragons, in addition to the Irish quartet of Leinster, Munster, Ulster and Connacht, according to the Sunday Telegraph.
"Rugby is at the heart of BT Sport, and so we talk to people all the time," a BT Sport spokesperson told Press Association Sport.
"We have helped to raise the profile of the club game in England by showing three live matches at weekends, and we plan to build on the successful start we have made."
BT Sport's fierce rivals Sky will broadcast live RaboDirect PRO12 games for the first time from next season, while BT Sport recently outbid Sky by landing a three-year £900million deal to screen UEFA Champions' League games.
And BT Sport's reported interest in expanding their rugby portfolio comes with the future of European Cup rugby for next season still shrouded by uncertainty.
Talks between the six nations comprising top-level European rugby - English, Welsh, Scottish, Irish, French and Italian - took place in Dublin on Friday, with further discussions planned for next month.
At the moment, England's 12 Aviva Premiership clubs will not play in European competition from next season after refusing to negotiate with European Rugby Cup. Premiership Rugby began serving a two-year notice period to quit ERC in June 2012.
Premiership Rugby have also seen their attempts to form a breakaway Rugby Champions Cup dashed, at least for the time being, and English clubs could face a season without European rugby next term, although there has also been intense speculation about a new Anglo-Welsh league being formed.
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- Newport Gwent Dragons
- BT Sport