Four men in a white Subaru vehicle stopped his car and forced him out before driving off with him, they said.
Simon Wilson, the bureau chief for the BBC in Jerusalem, said they had lost contact with Johnston but did not confirm he had been kidnapped.
Johnston, a Scot and British citizen, has been working as the Gaza correspondent for the state-funded British broadcaster for three years.
The British embassy said they were investigating the reported kidnap.
The last foreigner to be kidnapped in Gaza was a Peruvian photographer from the French news agency, Agence France-Presse, in early January.
The AFP photographer was released unharmed after nearly one week in captivity.