He clearly desperately wants to be PM but the only way he'd get there would be if Johnson suddenly resigned and Gove somehow won a leadership bid. But I can't see the general public voting for him.
He's tried that already. He can't even get past the bit where Conservative MPs like him enough to vote for him.
The first time he tried it, he was in a popularity contest with Liam Fox, Stephen Crabb, Andrea Leadsom and Theresa May. Liam Fox is a man with a talent for being utterly useless and had a glittering career of scandal behind him. Stephen Crabb was forced to leave the contest for personal reasons. And Gove was knocked out by May and Leadsom.
Let me stress this. Tory MPs thought that Leadsom was more trustworthy/credible/capable/insert your own reasoning than Gove to lead the country.
Then three years later, we go through the whole thing again.
This time, just about every Tory MP sees this as the Big Opportunity. So theres lots of candidates. Gove has to work hard to repair his reputation after the previous backstabbing issue. Gove, once again can't beat third place, losing out to the more credible Hunt and the incredibly popular (and who most thought had it in the bag from the off) Johnson.
I do note here, that Johnson was felt to have 'cleared his decks of personal scandals' by ditching his wife and making it public that he was shagging Carrie in the run up to the the 2019 leadership election / general election to make him more electable and for it not to ruin his chances by it all breaking at a more crucial moment.
Rumour has been that Johnson is fed up and wants out and won't do more than one term.
Gove has slowly been trying to detoxify his image and make himself look capable. But he is still hugely disliked within his own party ranks.
Its widely known that there has been three camps briefing against each other for months - the Cummings Camp, The Johnson Camp and the Gove Camp though the Gove camp has managed to not get as much heat / public awareness.
So yeah that is at the back of my mind. But he still has to convince the Tory MPs to get through stage 1 of the Tory Leadership Election and then he has to convince the Tory Party Membership.
Quite frankly, keeping Gove out of No10 does seem like a good reason to join the Party tbh