The 25th Amendment was still under consideration in case the president becomes more unstable in his last days of office, a source close to Mr Pence told CNN.
However there are concerns that using the amendment or pushing impeachment could encourage Mr Trump to act rashly thus putting the nation at even greater risk, the source added.
The 25th Amendment states that the vice president would take over if a sitting president is removed, resigns, or dies while in office. Invoking the amendment requires approval from the vice president and a majority of the president’s Cabinet.
Tensions had mounted between Mr Pence and Mr Trump in the days leading up to the US Capitol riots. The president had been pushing for Mr Pence to challenge the election certification in Congress, despite the Constitution giving the vice president no power to do so.
During his Wednesday ‘stop the steal’ rally, Mr Trump said he would be “very disappointed” in his vice president if he failed to act. According to reports, the president never checked in with Mr Pence during the riots which subsequently ensued at the US Capitol.
The pair have not spoken since the violence at the Capitol, CNN reports. Five people, including one Capitol Police officer, died in the riots.
Mr Trump has not publicly condemned the social media death threats against his vice president. The White House issued a statement on Saturday saying they “strongly condemn all calls to violence, including those against any member of this administration”.
One of the final tweets Mr Trump sent, before his account was permanently removed by Twitter, was to announce he would not be attending Mr Biden’s inauguration.
Meanwhile an aide to the vice president said Mr Pence would be attending the inauguration on 20 January, The New York Times reports.
Last week Mr Biden said he was glad to not have Mr Trump attend his inauguration given all that happened at the US Capitol. But he added that Mr Pence was “welcome” to the event and the vice president’s presence would help with the transition process.