The New York Times, quoting people familiar with the matter, said the hacking attempt appeared to be part of a wide-ranging intelligence collection effort by Chinese hackers.
The newspaper said it involved the infiltration of Verizon phone systems and investigators were seeking to determine whether any communications data was taken.
A person familiar with the matter said the campaign of Democratic candidate Kamala Harris and her running mate Tim Walz was also targeted.
Rich Young, a spokesman for Verizon, told AFP: "We are aware that a highly sophisticated nation-state actor has reportedly targeted several US telecommunications providers to gather intelligence."
The Times said the Trump campaign was informed this week that the former president and Vance were among a number of people inside and outside government whose phone numbers had been targeted.
Western cybersecurity experts believe the hacking was carried out by a Chinese group known as "Salt Typhoon," the newspaper said.
The Washington Post said the hacking campaign is believed to have compromised the phones of two staffers on the Trump-Vance campaign.
The FBI declined to comment on the reports that Trump's and Vance's phones had been targeted.
But the FBI and Cybersecurity and Infrastructure Security Agency (CISA) said in a joint statement that the "US government is investigating the unauthorized access to commercial telecommunications infrastructure by actors affiliated with the People's Republic of China."
"After the FBI identified specific malicious activity targeting the sector, the FBI and (CISA) immediately notified affected companies, rendered technical assistance, and rapidly shared information to assist other potential victims," they said.
"Agencies across the US Government are collaborating to aggressively mitigate this threat and are coordinating with our industry partners to strengthen cyber defenses across the commercial communications sector," they added.
The United States charged three Iranians last month with involvement in a hacking effort targeting the Trump campaign.
In 2016, a hack of Democratic National Committee emails -- blamed on Russians -- exposed internal party communications, including about candidate Hillary Clinton.
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