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The country's governing body said they had written to Cricket Australia that the winner of the two-Test series, opening in Brisbane on November 8, be awarded the Warne-Murali Trophy.
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"We have sent our proposals and have got a very good response," SLC secretary Kangadaran Mathivanan told AFP.
The SLC said it was confident that Cricket Australia would endorse the proposal to name the Trophy after two of the world's greatest spin bowlers.
Muralitharan is just nine wickets away from surpassing the retired Warne's world record of 708 Test wickets.
Sri Lanka last played a Test series in Australia in 2004 and the next contest may not take place before 2010.
If Australia accepts Sri Lanka's proposal, the Warne-Murali Trophy will be the latest addition to a series named after famous players. Australia's Tests against India are fought for the Border-Gavaskar Trophy, named after batting greats Allan Border and Sunil Gavaskar, while Australia and New Zealand compete for the Chappell-Hadlee o