The little boy has been at the centre of a lengthy legal struggle involving his parents, who wanted him to undergo a therapy trial in the US, and hospital specialists who said the treatment was experimental and would not help.
Chris Gard and Connie Yates are now spending the last days of their 10-month-old son's life with him, after being given more time before his life-support is turned off.
Charlie, who suffers from a rare genetic condition and has brain damage, is being cared for at Great Ormond Street Hospital (GOSH).
The US president tweeted on Monday: "If we can help little #CharlieGard, as per our friends in the U.K. and the Pope, we would be delighted to do so."
It comes after Pope Francis called for Charlie's parents to be allowed to "accompany and treat their child until the end".
Charlie's plight has touched people around the world and the family received donations totalling more than £1.3 million to take him to the US for therapy.
His parents, both in their 30s and from Bedfont, west London, asked European court judges in Strasbourg, France, to consider their case after judges in the High Court, Court of Appeal and Supreme Court in London ruled in favour of GOSH doctors.
But last week the European Court of Human Rights refused to intervene.
During a hearing at the High Court in April, Mr Justice Francis considered evidence from a specialist who would oversee any treatment Charlie had at a hospital in the US.
The specialist, who cannot be named for legal reasons, said therapy would provide a "small chance" of a meaningful improvement in Charlie's brain function.
He told the court via a telephone link from America: ''It may be a treatment, but not a cure.
"(Charlie) may be able to interact. To smile. To look at objects.''
In a statement released on Sunday, the Vatican press office said the pope "is following with affection and sadness the case of little Charlie Gard and expresses his closeness to his parents.
"For this he prays that their wish to accompany and treat their child until the end is not neglected".
On Friday, a picture of the couple sleeping on either side of their son in hospital was posted on their Twitter account alongside the hashtags #jesuisCharlieGard #charliesfight #letcharliegohome.
The couple released an emotional video a day earlier saying they had been told Charlie would die on Friday.
They said they had been denied their final wish to be able to take their son home to die and felt "let down" after losing their legal fight.
The hospital later confirmed it was "putting plans in place for his care".
On Sunday, campaigners carrying a banner that said "It's Murder" gathered outside Buckingham Palace to protest against the court's decision.
Other posters with pictures of Charlie said "Where there's life, there's hope" and "parental rights".
Prime Minister Theresa May's official spokesman said Downing Street was aware of the president's tweet about Charlie, but made no further response other than to say: "All our thoughts are with him and with his family."