Tate has slotted well into the Leeds back line after arriving last month, but his loan is set to expire in the next few days and manager Neil Warnock wants to keep the experienced centre-back at Elland Road.
The 30-year-old has slipped down the pecking order at Swansea and Laudrup is willing to let him stay in Yorkshire until the summer.
"Alan is doing well and I think he'll continue there for the rest of the season," he told the Yorkshire Evening Post.
Reports suggested that Leeds had been willing to offer £250,000 for Tate, who is under contract with Swansea until 2015, but a temporary deal appears more likely.
A longer loan for Tate will be further good news for Warnock, who has tied down El-Hadji Diouf on a fresh deal and agreed the permanent signing of fellow winger Ryan Hall from Southend.
Leeds are also expected to open talks about extending the loan spells of Stoke midfielder Michael Tonge and West Brom winger Jerome Thomas.