The current WBC champion revealed he is willing to defend his crown against fellow countryman Anthony Joshua, who faced an agonising second defeat against Usyk last month.
In a video posted on social media, Fury said: "I think you all heard that I will be fighting soon, within the next few months. I think before I announce an opponent, I need to do this just in case.
"Anthony Joshua, I know you've just lost a fight to Usyk and you're beltless at the moment. I'd like to give you an opportunity to fight me for the WBC heavyweight championship of the world and the lineal championship in the next few months.
"You're coming off a 12-round fight so you're match fit, you're ready. I'm giving you a few months' notice.
"If you're interested, I'll send you a date over and we can rumble. A battle of Britain for the WBC championship of the world; if you're not interested, let me know and I'll select another opponent."
Fury hinted at retirement after knocking out Dillian Whyte at Wembley stadium last April.
However, the Brit has since teased a number of comeback opportunities and appears intent on returning to the ring this year.
The 34-year-old made a recent appearance on WWE's Clash of the Castle event, where he revealed in a post-show press conference that a fight will be announced this week.