In an interview with German business daily Handelsblatt, Kimmitt said he was "very pleased" that the European Union had frozen the assets of Iran's largest lender, Bank Melli.
"The U.S. government has made clear it would like to continue the diplomatic path led by the European," Kimmitt told the paper.
"If Iran halts uranium enrichment all offers already made are on the table," he added.
"If not, we must intensify diplomatic efforts and consult about new sanctions."
Western nations have led an effort to isolate Iran, resulting in three rounds of U.N. sanctions, because Tehran has refused to halt work they fear is aimed at making atomic bombs.
Iran says it will not halt work it insists has peaceful aims but has yet to give a formal response to a package of trade and other incentives offered by world powers for Tehran to stop.
Kimmitt said so far three big Iranian state-owned banks had been hit by sanctions.
"I think that new sanctions would affect further Iranian banks," he said.
(Reporting by Iain Rogers, editing by Mary Gabriel)
