Ms Forster is the sister of former prime minister Tony Abbott.
Christine Forster, the sister of Tony Abbott, is considering running in the NSW seat of Wentworth after Malcolm Turnbull vacated it.
A spokesperson for Ms Forster confirmed to SBS News on Friday afternoon that she is considering running but declined to provide further details.
Ms Forster is a Liberal councillor in the City of Sydney and has been a vocal advocate of same-sex marriage.
Mr Turnbull indicated he would resign after the spill, triggering a by-election in the seat.
Scott Morrison was announced as Australia's 30th prime minister on Friday after being elected Liberal leader in a tense party room meeting.
It is understood Mr Morrison - who has served as treasurer since 2015 - won the leadership content 45 to 40 votes.
Ms Forster said she felt "despair" at how the latest leadership spill unfolded.
"There’s nothing that can describe the despair that blue-blooded Liberals like me feel watching what’s going on in our capital," she said in a message posted on Twitter.
She defended her brother Tony Abbott's record as prime minister, saying he did unite Liberals MPs.
"Actually, the Liberal party between 2009 and 2015 was united as a “broad church” under @TonyAbbottMHR," she tweeted.
In the last decade, Australia has had six prime ministers: Kevin Rudd (twice: 2007-10; 2013), Julia Gillard, Tony Abbott, Malcolm Turnbull and Scott Morrison.