Sheikh Salman Humoud Al-Sabah's move comes after around 30 lawmakers of the 50-member house announced their support for the no-confidence motion filed after MPs grilled the minister last week.
Three opposition lawmakers questioned Sheikh Salman on Tuesday over a failure to lift a 15-month international ban on Kuwaiti sports and over alleged administrative and financial violations.
They blamed the minister for causing the ban imposed on the Gulf state by the International Olympic Committee, the world football body FIFA and several sports federations.
FIFA and IOC imposed the suspension in October 2015, for the second time since 2010, over alleged government meddling in sports.
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"An official announcement of the resignation is expected to be made on Tuesday," a senior government source told AFP, requesting anonymity.
This week's confidence vote needed the support of just 25 MPs to pass. Under Kuwaiti law, approving the no-confidence motion means automatic dismissal.
MPs Waleed al-Tabtabai, Abdulwahab al-Babtain and Alhumaidi al-Subei, who grilled Sheikh Salman, said in statements today the minister had resigned and vowed to cooperate with his successor.
It has also urged the emirate to withdraw all cases it has filed in Swiss courts against international sports bodies challenging the suspension's legality.
As a result of the suspension, Kuwait was barred in early January from taking part in qualifiers for the 2019 Asian Cup.
The wealthy emirate has already missed out on the Rio 2016 Olympic Games and qualification for the 2018 World Cup.
Analysts say the crisis was partly caused by a political struggle involving senior ruling family members and politicians.
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