Chris Bryant, the Labour MP for Rhondda, referred to recent statements by Mr Musk on Twitter and said there were questions about reported cuts to his Starlink system in Ukraine.
Mr Bryant has called for sanctions to be placed on several people since Russia invaded Ukraine, including former Chelsea owner Roman Abramovich, who was then penalised by the UK.
As members of the House of Commons asked a defence minister about the situation in Ukraine, Mr Bryant said: “Can I ask him about Elon Musk?
“Because he seems to be playing a double game at the moment, and his tweet earlier on this week I think was profoundly unhelpful.
“There are also questions about why there have been outages of the Starlink system which may have … made bigger difficulties for Ukraine.
“Is there a moment at which we might have to consider sanctioning Elon Musk?”
Elon Musk's 2022 - in pictures
Defence Procurement Secretary Alec Shelbrooke appeared to be surprised by the question.
“Sanctions remain under review at all times and everything will be taken into consideration in the round," Mr Shelbrooke said.
“I think we must always make sure we’re well aware of what all the facts are, not just react to social media, and then those things can be looked at and whether any sanctions would be appropriate or not.”
Starlink is a satellite-based internet system created by Mr Musk’s SpaceX.
The Financial Times reported that Ukrainian troops on the frontline have recently reported cuts to their Starlink communication devices.
Some of the devices being used in Ukraine were donated by SpaceX, while others have reportedly been bought and supplied to Ukraine by the US government and others.
The FT quoted a Ukrainian government official who said that the blackouts led to a “catastrophic” loss of communications in recent weeks.
According to the report, Mr Musk and SpaceX did not respond to requests for comment, but he later said on Twitter: “As for what’s happening on the battlefield, that’s classified.”
Twitter investors backing Musk - in pictures
Mr Musk, the chief executive of Tesla, last week tweeted that to reach peace, Russia should be allowed to keep the Crimean Peninsula it seized in 2014.
He also said Ukraine should adopt a neutral status, dropping a bid to join Nato after Russia’s partial mobilisation of reservists.
Mr Musk also crossed red lines for Ukraine and its supporters by suggesting that four regions Russia is moving to annex after “referendums” denounced by the West as a sham should hold repeat votes organised by the UN.
He also launched a Twitter poll asking whether “the will of the people” should decide if seized regions remained part of Ukraine or became part of Russia.
In a sarcastic response, Ukrainian President Volodymyr Zelenskyy posted a Twitter poll of his own, asking “Which Elon Musk do you like more?” — “One who supports Ukraine” or “One who supports Russia”?