The Mirror says Lambert has urged Aston Villa owner Randy Lerner to cough up for Ki.
Lambert wants Swansea's South Korean international to anchor his midfield this season.
Ki has only 12 months left on his contract at The Liberty Stadium but the Swans have refused to sell for a knock-down fee.
Now Villa's Scottish boss is urging wantaway owner Lerner to dig deeper to get his man, valued at £4million by the Welsh club.
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