The group claimed on Friday that the US aid worker had been killed by a Jordanian air attack.
A White House spokesperson has said her parents received a private message from IS over the weekend, which has since been authenticated.
Kayla Mueller's family have also released a letter she wrote to them from captivity, expressing her faith in God.
Kayla's message reads as follows:
"Everyone, If you are receiving this letter it means I am still detained but my cell mates (starting from 11/2/2014) have been released. I have asked them to contact you + send you this letter.
"It's hard to know what to say. Please know that I am in a safe location, comple4ly unharmed + healthy (put on weight in fact); I have been treated w/ the utmost respect + kindness. I wanted to write you all a well thought out letter (but I didn't know if my cell mates would be leaving in the coming days or the coming months restricting my time but primarily) I could only but write the letter a paragraph at a time, just the thought of you all sends me into a fit of tears.
"If you could say I have 'suffered' at all throughout this whole experience it is only in knowing how much suffering I have put you all through; I will never ask you to forgive me as I do not deserve forgiveness.
"I remember mom always telling me that all in all in the end the only one you really have is God. I have come to a place in experience where, in every sense of the word, I have surrendered myself to our creator b/c literally there was no else.... + by God + by your prayers I have felt tenderly cradled in freefall.
"I have been shown in darkness, light + have learned that even in prison, one can be free.
"This should never have become your burden. I have asked these women to support you; please seek their advice. If you have not done so already, [REDACTED] can contact [REDACTED] who may have a certain level of experience with these people. None of us could have known it would be this long but know I am also fighting from my side in the ways I am able + I have a lot of fight left inside of me.
"I am not breaking down + I will not give in no matter how long it takes. I wrote a song some months ago that says, 'The part of me that pains the most also gets me out of bed, w/out your hope there would be nothing left...' aka-The thought of your pain is the source of my own, simultaneously the hope of our reunion is the source of my strength. Please be patient give your pain to God. I know you would want me to remain strong. That is exactly what I am doing. Do not fear for me, continue to pray as will I + by God's will we will be together soon.
"All my everything,