The report came just hours before tariff exemptions were due to expire at midnight for the EU, Canada, Mexico and others, after which the major exporters would face punishing import duties of 25 percent on steel and 10 percent on aluminum.
The postponement would effectively leave the exemptions in place.
Trump announced the tariffs on March 8, but exempted the European Union later that month.
The Trump administration has justified the tariffs — aimed primarily at Chinese overproduction — on national security grounds and told trading partners they must make concessions to win permanent exemptions.
The EU has insisted it will not negotiate without first obtaining a permanent exemption.
The US official, speaking to the Journal on condition of anonymity, also said that the Trump administration would announce later Monday it had reached a deal to exempt South Korea from the tariffs.
Trump is working with South Korean President Moon Jae-in ahead of the US leader’s expected summit with North Korea’s Kim Jong Un in the coming weeks.
Last month, the government in Seoul said it had made concessions to secure a revised trade deal with Washington and escape its steel duties.
Extensions for Canada and Mexico had been expected, as Washington, Mexico City and Ottawa work on revamping the North American Free Trade Agreement.