The brothers acknowledged that the current level 4 lockdown amid the Delta outbreak in New Zealand had put "additional stress and financial pressure" on many Kiwis.
He continued: "So any manageable donations towards Francis' funeral and repatriation costs will be greatly appreciated and help our family during this difficult time."
They also paid tribute to their brother, writing: "Francis was an energetic force and much-loved brother and son.
"He was a well-respected member of the acting community and found a supportive and endearing family community in Sydney. His smile and energetic presence will be sorely missed by those lucky enough to have known him."
The 33-year-old actor starred in Shortland Street and Spartacus: Blood and Sand. Several outlets, including the LGBTQ website Queer Screen Australia and the Daily Mail posted the tragic news of his death.
Mossman, 33, died on Saturday in a suspected suicide.
"Queer Screen was saddened to hear of the passing of actor, Frankie Mossman," a post on Queer Screen's Facebook page read.
"Moving from Auckland to Sydney in 2012, Frankie was a well-known member of our LGBTIQ+ community, appearing in numerous NZ and Australian productions including the Sydney-made gay web series, The Horizon.
"Frankie was a great friend of Queer Screen, starring in our MGFF16 trailer and was always an enthusiastic audience member at our festivals. Our sincere condolences to all his friends and family. May he Rest In Peace."
In 2020, Mossman starred in the short film Dis-Connect in 2020. He starred as Taylor in Shortland Street in the mid-2000s.
His friends and followers paid tribute to the actor in the comments of his social media posts.
Colin Mathura-Jeffree wrote: "Just so so sad to hear the news. Strangely you have been in my thoughts these past 3 days. Fly High ..."
Australian dancer and entrepreneur Peter Oxford wrote: "I am so sorry to hear you have left us. I was only talking to you this week about going back to NZ."
Where to get help:
• Lifeline: 0800 543 354 (available 24/7)
• Suicide Crisis Helpline: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• Youthline: 0800 376 633 or text 234 (available 24/7)
• Kidsline: 0800 543 754 (available 24/7)
• Whatsup: 0800 942 8787 (12pm to 11pm)
• Depression helpline: 0800 111 757 or text 4202 (available 24/7)
• Anxiety helpline: 0800 269 4389 (0800 ANXIETY) (available 24/7)
• Rainbow Youth: (09) 376 4155
If it is an emergency and you feel like you or someone else is at risk, call 111.