Mr Johns was transferred to a specialist burns unit in Swansea after a blaze broke out at his home in Bristol around three-weeks-ago.
In a statement his family said Mr Johns is now "off the ventilator and is conscious".
"Jeff has made some good progress with his recovery over the weekend," they said.
"Jeff is beginning to be able to receive the wonderful messages of love, support and goodwill - thank you."
Bristol band Idles dedicated a song to the well-known concert-goer during their set at the Glastonbury Festival.
And on Monday, during an open-air concert on Bristol's Harbourside, The War on Drugs' lead singer Adam Granduciel paid tribute to Mr Johns, and said the band were thinking of him.