Denmark is sending the 139-meter Iver Huitfeldt frigate to the area as part of Operation Prosperity Guardian formed last month to protect merchant vessels.
US and British warplanes, ships and submarines have launched dozens of air strikes across Yemen against Houthi forces in retaliation.
"While en route, both ships reported seeing explosions close by and the US Navy accompaniment also intercepted multiple projectiles," Maersk said in a statement.
White House National Security spokesman John Kirby said the US is taking away the Houthis' capability with "each and every strike."
Reuters has tracked at least 22 other oil tankers to have diverted or paused since last Friday's warning from the US-led Combined Maritime Forces (CMF) in Bahrain.
Although the warring sides agreed to a draft peace roadmap to end Yemen's civil war, Houthi attacks on ships in the Red Sea and US-UK airstrikes have analysts asking if the peace plan will go ahead.
The UN Security Council on Wednesday demanded Yemen's Houthis immediately end attacks on ships in the Red Sea.
"The threat to navigational rights and freedoms in the Red Sea is a global challenge that necessitates a global response," US Ambassador Linda Thomas-Greenfield said.
Shapps warned that “The Houthis will bear (the) responsibility of the consequences should they ignore these warnings.”
"We haven't really seen the interruption to tanker traffic that everyone was expecting," said Michelle Wiese Bockmann, a shipping analyst at Lloyd's List.
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