Jacqui Smith says she has fully complied with parliamentary rules
Jacqui Smith should explain her conduct in claiming "second home" expenses for the house she shares with her husband and children, the Conservatives say.
She can claim up to £24,000 a year for her family home in Redditch because she has told the Commons her sister's house in London is her main residence.
The home secretary has denied doing anything wrong, saying she has "fully abided" with expenses rules.
But Tory leader David Cameron said she "may have questions to answer".
Speaking in London, Mr Cameron said the reports - in the Mail on Sunday - did not look "very good".
And a formal complaint has been made to the parliamentary standards watchdog about Ms Smith's behaviour.
The Standards Commissioner said it had received the complaint - from pressure group Centre for Open Politics - but made clear it had not decided whether to look into the issue.
According to the newspaper, Ms Smith has claimed more than £116,000 in second home expenses, known as the Additional Costs Allowance, over several years.
Number 10 backing
The newspaper said Ms Smith stays at her sister's home when she is London, normally between Monday and Thursday.
The allowance is to help cover the cost of MPs having to have a second home - or place to stay - in London if their constituency is too far away to allow commuting to Parliament.
It can cover mortgage interest, fuel bills and things like a new kitchen, bathroom or even the purchase of items such as flat screen televisions and fridges.
The rules state that the main residence is where the MP "spends more nights than any other".
Under the rules, which have been in the spotlight over the past year, Parliament leaves it up to MPs to decide whether their London or constituency home is their "second home".
A spokeswoman for Ms Smith has said she spends most of her time in London on government business and has "full approval" from the Commons Fees Office for associated expenses relating to her second home in the West Midlands.
Ms Smith pays more than a "peppercorn" rent at her sister's London home, the spokeswoman added.
A No 10 spokesman backed Ms Smith on Monday, pointing to the home secretary's statement on the matter.
But Mr Cameron said it was right that Ms Smith should be expected to justify her arrangements as "reasonable".
"I think she may have some questions to answer when you look at all the rules that govern this.
"There are rules and you have to meet both the letter and the spirit [of the rules] and explain in a reasonable way 'this is the arrangement I have and I think it is a reasonable one'."
The Lib Dems have said they will ask the Commons authorities to look again at the designation of first and second homes.
The rules over designating a main residence were recently clarified after a complaint about Ms Smith's cabinet colleagues Ed Balls and Yvette Cooper.
They were cleared in October of any wrong-doing over earmarking their constituency home in Yorkshire as their main residence rather than their London home.
In his ruling on that case standards commissioner John Lyon said it was difficult to determine which home MPs should claim as their main home.
Mr Lyon concluded his report by saying that the "normal criteria" that the main home should be where an MP spends more nights than anywhere else should remain the "reasonable general test".