Corbyn has indicated that he could stay on as Labour leader, even if the party suffers a defeat at the next general election.
Mr Corbyn conceded on Thursday evening at an event in Birmingham that the party will have to win over people who have been "tempted to vote Tory" to have a chance of forming a Labour government. "We can't afford to spend another moment fighting each other".
In August, Owen Smith gained a nominational vote by the Jewish Labour movement, receiving 92% while Corbyn received just 4%.
He added: "Simply opposing Tory policies will never be enough to help the people we exist to support". As a result, many of Corbyn's shadow cabinet members resigned and publicly stated they had no confidence in his leadership.
Labour had campaigned for Britain to remain in the European Union (EU) but Corbyn - the extreme left-wing MP who was elected leader of the main Opposition party in a landslide victory previous year - was believed to be more inclined towards Brexit despite campaigning for remain.
Elsewhere in the debate, Mr Smith berated Mr Corbyn for his "ineffectual leadership" in the Shadow Cabinet and for failing to address Brexit at the first Prime Minister's Questions after the European Union referendum.
But he admitted during the party's latest leadership debate that Labour will have to appeal to people who have previously considered voting for the Conservatives.
The Scottish Labour leader's office said Dugdale had no plans to meet either Corbyn or Smith before or after their sole Scottish clash at Glasgow's SECC on Thursday. However, preaching to the converted and appealing to the general public are two fundamentally different things.
In a boost to MP Owen Smith, who is challenging Corbyn for the Labour leadership, Khan broke his silence over who he was backing in the contest.
Gavin Callaghan, leader of Basildon Council Labour Group, said: "He is coming to meet residents and members from across Essex".
Mr Smith said he was "honoured" to receive Mr Khan's support.
Mr Smith drew jeers from some when he referred to "170 socialist MPs" who lack confidence in Mr Corbyn's leadership.
Mr Corbyn has been criticised for failing to do enough to tackle abuse and allegations of anti-Semitism within the party. This nomination came after a Jewish donor to the British Labour Party, Michael Foster, likened Corbyn and his inner circle to "Nazi Stormtroopers'".
Members took part in a secret ballot following a 30-minute discussion of candidates Mr Corbyn and rival Owen Smith. Corbyn also came under fire for running a lackluster campaign in favor of the Remain movement.