There is no consensus, and we can never say for certain, and some authors will outright say Jochi was not Chinggis' son, such as Timothy May, who notes that Jochi means "guest," while some such as Christopher Atwood and Igor de Rachewiltz seem to be ambivalent on the matter while Anne Broadbridge implies that he was.

And, as almost every date in Chinggis' life before 1200 are all contentious, we can't say for certain how long it was. The Secret History of the Mongols says essentially nothing on the matter of his parentage (it does not the Jochi-Chagatai enmity in regards to it however), and I believe (but I'd have to reread them) that Rashid al-Din and Juvaini say Jochi was his son and I'd it's entirely possibly even Borte herself didn't know the truth of the matter.

However, I would say the historiography probably leans more towards Jochi not being his son: I believe it was in Frank McLynn's biography of Chinggis that he suggested that the uncertainty was manufactured to 'protect' Chinggis' masculinity, essentially.

But, Mongol adoption laws meant that an adopted son could be considered as much as son as any of biological sons he had. See for example the concept of Anda, blood brother, which while not adoption was considered legally binding in a similar manner. As well, Toghrul's (Ong Khan) adoption of Temujin in Winter 1202/1203 meant that Temujin could have been considered his heir, causing Toghrul's son Senggum Ilkha to encourage the split between them later that year.

With that in mind, that may be why everyone found Chagatai's hatred of Jochi so strange: even if biologically Jochi wasn't Chinggis' son, they still had the same mother and as Chinggis considered him his son anyways, that was just as strong legally.

As well, while it is certainly interesting that the Jochids don't seem to have made a play for the position of Great Khan (perhaps in an unspoken recognition of this uncertainty, according to Peter Jackson? understandings that Chagatai's descendants wouldn't allow it?) their right to be Princes was not contested. Ultimately, the truth can't be known

Sources:

Anne Broadbridge, "Marriage, family and Politics: The Ilkhanid-Oirat Connection." Journal of the Royal Asiatic Society, 2016.

Frank McLynn, "Genghis Khan: The Man who Conquered the World," The Bodley Head, 2015.

Peter Jackson, "The Mongols and the Islamic World," Yale University Press, 2017.

Timothy May, "The Mongol Conquests in World History," Reaktion Books, 2012.

Igor de Rachewiltz. The Secret History of the Mongols: A Mongolian Epic Chronicle of the Thirteenth Century. Translated by Igor de Rachewiltz. Edited by John C. Street. University of Wisconsin: Madison, 2015. http://cedar.wwu.edu/cgi/viewcontent.cgi?article=1003&context=cedarbooks

EDITED: formatting on sources list