0748 GMT April 20, 2018
The British Boxing Board of Control suspended Fury's license in 2016 owing to "anti-doping and medical issues", BBC reported.
Fury, 29, accepted a backdated two-year anti-doping ban in December and is working toward a return in 2018.
He tweeted on Wednesday that he would be "applying for my license today" and would be back in the ring in April.
But the BBBofC has not spoken to Fury and insists the process "really can't move on" without medical records.
Fury called out Joshua in a series of tweets which read, "As a sporting challenge I Lay Down the challenge for Anthony Joshua to fight me in my first fight back in over 2.5 years!"
"I'm the best heavyweight on the planet come prove I'm not? The ball is in your court don't let your fans down CHAMP."
"We have a number of reports already in from when he was suspended," BBBofC general secretary Robert Smith said.
"We will need up-to-date ones and then pass them on to our doctors and consultants. They will tell me if they need more but at present we are nowhere near that.
"I have spoken to Mick Hennessy [Fury's promoter], advising he'd like to get on with this as soon as possible and he did say he'd come back to me but unfortunately at this time he hasn't so it hasn't moved on yet."
Smith's comments come as Fury frequently shares training videos with fans on Instagram, reportedly losing four stone in order to move closer to his fighting weight.
The Manchester-born fighter has not competed since claiming the IBF, WBA and WBO titles by beating Wladimir Klitschko in 2015.
On Sunday, Fury tweeted he was looking forward to returning to the ring "in late April", while 24 hours earlier he tweeted IBF and WBA world heavyweight champion Anthony Joshua stating he was "ready for this summer".
In October, with Fury vocal over the lengthy nature of his UKAD case, he criticized the BBBofC and said he would not seek a new license from them.
He could look to other bodies but the BBBofC would need to be satisfied with that organization’s credibility in order for Fury to fight in the UK.
"Tyson has a license with us that has been suspended," Smith added. "Until that is lifted, we really can't move on. If he is to fight here under someone else's license, we have to make sure that organization is fit and proper."