Iraq, Pakistan, Turkey, Afghanistan, Armenia, Jordan, Saudi Arabia and Kuwait have imposed temporary bans on people travelling to and from Iran.
"We have decided to shut the land border temporarily after an increase in the number of cases in our neighbour Iran," Turkish Health Minister Fahrettin Koca said on Sunday.
Pakistan has closed its land border with Iran, while Afghanistan has suspended all travel to Iran.
"To prevent the spread of the novel #coronavirus and protect the public, Afghanistan suspends all passenger movement (air and ground) to and from Iran," the office of the National Security Council of Afghanistan said in a statement posted on Twitter.
Saudi Arabia has said that anyone travelling from Iran must wait for at least 14 days before entering the kingdom to prevent the spread of the virus to the Muslim pilgrimage sites of Mecca and Medina.
Armenia has become the latest country to close its border with Iran. According to Armenia's Prime Minister Nikol Pashinyan, his country is closing the border for two weeks and suspending air traffic amid growing alarm over a rise in the number of coronavirus cases in Iran.