Second row Beirne, 25, has been linked with Munster after playing a pivotal role in the Welsh region's championship victory last season.
The lock joined from Leinster and is being targeted by the team Scarlets beat in the Pro12 final in May.
"We've spoken to his agent and to Tadgh and offered him an extension on his contract," said Pivac.
"He has a bit of a dilemma in that he is playing so well that all of a sudden there are some noises coming out of Ireland.
"I don't think they wanted to know him 12 months ago, and that's why he came here to the Scarlets to get some regular rugby."
Beirne played 24 games in the 2016-17 season and scored a try as Scarlets beat Munster 46-22 in the Pro12 final at Dublin's Aviva Stadium.
Pivac believes any decision will be based on Beirne's Ireland international ambitions.
"He's one we will continue to talk to and he has another big season coming up here," said Pivac.
"If he doesn't back up his performances from last season then there won't be any noises coming out of Ireland.
"He wants to play for Ireland and who can blame him? We'll wait and see how that one pans out."
Beirne is one of more than 20 Scarlets players out of contract at the end of the 2017-18 season, including Wales internationals Scott Williams, Gareth Davies, Jake Ball and Olympics Sevens silver medallist James Davies.
Ball is on a National Dual Contract with the Welsh Rugby Union which is a similar deal to Scarlets centre Jonathan Davies.
Talks about extending Davies' contract will also continue as the Wales centre was named the British and Irish Lions player of the series against New Zealand.
"We're well down the track," said Pivac.
"Before I went to New Zealand to watch the Lions we had discussions and we've been talking to agents in the background for some time.
"We've identified the players we want to keep and we've touched base with them early.
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"I'd like to think we'd get them done as soon as possible.
"We've got a good squad of guys. It's taken three years to get to this stage and we want to keep building.
"We'd like to keep the nucleus of this squad.
"There is always going to be turnover - that's professional rugby - but we've got a good core of players who we are enjoying working with.
"We don't want to break that up."
Tadgh Beirne has been offered a contract extension, says Scarlets head coach Wayne Pivac.