The three-time Grand Slam champion said he was still 'apprehensive' but hopes tennis can follow other sports back into action after the long lay-off.
It will be screened live on Amazon Prime and aims to raise at least 100,000 pound (USD 122,000) for the state-run NHS.
Murray is back on the practice court after being unable to hit for three months because of what was initially diagnosed as a mild bone bruise.
Andy Murray just returned to training after being out of action since November due to bruising on his pelvic bone.
Bob Bryan had a major hip operation in 2018 but bounced back to continue his hugely successful doubles partnership with twin Mike.
Murray's Grand Slam comeback at next month's Australian Open, which starts on January 20, is apparently not in doubt, however.
Murray, now ranked a lowly 243 in the world, had the chance to wrap up the quarter-final when he had a match point in the second set tiebreak.
Fognini collapsed in fits of laughter at the net after one umpiring decision and then had an animated exchange with the official in Italian.
In his opener Murray claimed his biggest post-surgery scalp yet, defeating 13th-ranked US Open semi-finalist Matteo Berrettini.
Murray, 32, won his opening match at the Shanghai Masters on Monday, beating Argentine qualifier Juan Ignacio Londero in three sets, following a quarter-final appearance in Beijing last week.
Scotsman Murray, who is on the comeback trail after career-saving hip surgery, reached the quarter-finals of the China Open where he lost to eventual winner Dominic Thiem.
Barty now faces Kiki Bertens, after the Dutch eighth seed defeated No. 3 seed Elina Svitolina 7-6 (6), 6-2.
The former world number one, who was crowned Wimbledon champion in 2013 and 2016, is making slow progress in his return from hip surgery in January.
Wimbledon starts on Monday and Murray, who hasn't publicly announced he is fit enough to play, is due to face the media in a press conference on Saturday.
Andy Murray slipped to a 6-4, 6-4 defeat to British compatriot Kyle Edmund at Eastbourne on Wednesday to put his participation at Wimbledon in fresh doubt.