Saints have improved greatly of late, having lost just two of their previous nine Barclays Premier League fixtures heading into their match with Arsenal.
The north Londoners have taken Alex Oxlade-Chamberlain and Theo Walcott from the south coast club in recent years and have been linked with a move for 17-year-old academy graduate Shaw.
Manchester City and Chelsea have also reportedly shown interest in the left-back, although Adkins does not expect him to be going anywhere.
"I've not read any of the newspapers," the Saints boss said.
"I think Luke gets linked every week with Arsenal, doesn't he?
"Luke is under contract, he is playing football for us in the Premier League, which is important.
"We are not looking to let him go anywhere if anyone wants to come in for any of our players, by all means speak to the chairman. He does the deals and all the best with that."
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- Theo Walcott
- Barclays Premier League