The couple announced the end of their relationship after nearly four years together earlier this week.
© Rex Features / Brianne Makin/Newspix
While busy work schedules were already putting a strain on the relationship, Smith's Christmas present for Minogue was the final straw, The Mirror claims.
Model Smith is said to have given Minogue a new passport holder as hers had become "tatty from so much travelling". Friends have hinted that the gift was a small dig at how much she had been away from home.
"There have been lots of problems and the pair were living increasingly separate lives," a source is quoted as saying.
"Dannii is a workaholic, Kris wanted her to slow down and enjoy life a bit more, but you don't tell a Minogue that. Dannii is a perfectionist - she lists everything around the house like chores on little Post-it notes. They are everywhere, and when things are tense this kind of thing can cause a row."
The couple welcomed son Ethan in July 2010.
Minogue recently said that she would return to The X Factor if asked by Simon Cowell.
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