Spokespeople for Andreessen and Silver Lake declined to comment, while the reports noted they did not know when or if a deal would take place.
Shares were as much as 8% higher at US$18.25, and were around 7.5% up after a few hours of trading on Monday to US$18.
Twitter’s shares declined 35% last year as growth stalled, and the company is in the midst of a major overhaul to correct this.
The firm announced co-founder Jack Dorsey as its permanent chief executive in October, and he was quick to make his presence felt.
In January, four executives left, including the product and engineering chief.
The social media company also named a new chief marketing officer, Leslie Berland, formerly of American Express, and is expected to appoint two new board members soon.