SNP MP Tommy Sheppard said the independence campaign should be “parked” until Brexit negotiations are over.
Ms Sturgeon said independence was a factor after her party lost 21 of its 56 Westminster seats in the General Election, and she vowed to “reflect on the result”.
Mr Sheppard told the Sunday Herald newspaper: “Amidst the current chaos in Westminster it seems certain that a hard Brexit is now off the table, and the possibility of bespoke solutions for nations and regions is growing.
“It follows, therefore, that it is now an option to wait until the Brexit negotiations conclude before forming a view on whether the extent of change justifies a second independence referendum as a result. This would mean that whilst a second referendum remains an option, the timetable gets parked.”
Mr Sheppard put himself forward for selection as the party’s Westminster leader this week before withdrawing citing “lack of support”.
Meanwhile, former SNP political adviser Alex Bell said the party should “shut up about indyref2”.
He wrote in the Sunday Times: “Sturgeon should do the right thing by the nation and commit to real improvement in health and education, knowing the benefits may not be evident until after the 2020 election, and do the right thing by independence by explaining why short to mid-term risk is worth taking for long-term gain.
“Until that time, shut up about indyref2. The UK is in genuine crisis and there’s no loss of face in letting that settle until some undetermined date. And shut up about Brexit - play a part in softening it, but stop making it seem like the business of Holyrood to fix.”
The latest interventions come after former justice secretary Kenny MacAskill said the SNP should “abandon” its current push for a further independence referendum as “neither the mood is there for it, nor the support in existence to win it”.
He said on Saturday that the campaign should be sidelined following the party’s performance in the General Election, which he said cannot be “ignored or explained away simply as unionist tactical voting”.