David Tebbutt, 58, and his wife Judith, 56, were on holiday at the remote Kiwayu Safari Village, close to the border with Somalia, when the attack happened in the early hours of Sunday morning.
District Commissioner Stephen Ikua is quoted by The Times as saying a local man has been arrested on suspicion of helping coordinate the attack and is being held at a police station on the nearby island of Lamu.
Meanwhile, the Kenyan military has joined the hunt for Mrs Tebbutt.
Officials have said the Tebbutts were attacked on the first night of their stay at the resort, with reports suggesting bandits broke into their accommodation.

Daivd Tebbutt has been described as a 'lovely' and 'caring' man
There are suspicions that the gang, thought to be from Somalia, used a speedboat to get away from the isolated island resort, reportedly favoured by artist Tracey Emin, actress Imelda Staunton and Sir Mick Jagger.
The couple had come from visiting the Masai Mara reserve and were the resort's only guests.
Police Commissioner Mathew Iteere said the bandits could gain access to the couple's accommodation, which had a piece of cloth as a door, "so easily".
He told a press conference it was possible that Mr Tebbutt had "resisted", which may have been why he was shot.
They might want to use their hostage for negotiating leverage with foreign powers.
Pen Point's security editor makinde bayo
He said if the attackers are hoping for a ransom for Mrs Tebbutt, it was likely they will get in contact.
There has been speculation that the kidnappers could be from an al Qaeda-linked insurgent group called al Shabab, which controls much of southern Somalia.
Mr Tebbutt, from Bishop's Stortford, Hertfordshire, worked for publisher Faber & Faber and was a member of the Book Trade Charity, which offers support and grants to those in the book trade.
Chief executive David Hicks, who met him more than 10 years ago, said the best way to describe Mr Tebbutt was "all the dedicated words you want".
Sky's Security Editor On Possible Kidnappers.
"He was a lovely chap, he was on the grants committee. He was a very caring person and very concerned about the people that we were supporting financially."
He said he knew Mr Tebbutt, who was also a member of literary dinner club the Society of Bookmen, had lived and worked in Africa previously, although he did not know the details, but he did not know his wife.
The Foreign Office said a team has been deployed to the area from the High Commission in Nairobi and is "offering all possible support to the family of those involved".
A spokesman said: "We are working to secure the safe and swift release of the British national who has been kidnapped and ask those involved to show compassion and release the individual immediately."

The Kiwayu Safari Village is near the border with Somalia
It warned against "all but essential travel to within 30km of Kenya's border with Somalia".
Its website says: "There have been previous attacks by Somali militia into Kenya. Three aid workers were kidnapped in July 2009, and two western nuns in November 2008."
It also warns against piracy, referring to the kidnapping of two British nationals in October 2009 as they sailed from the Seychelles to Tanzania in notoriously dangerous waters.
Retired couple Paul and Rachel Chandler, from Tunbridge Wells in Kent, spent 388 days in captivity until they were released last November after a ransom of up to £620,000 was paid.
"Sorry our website is unavailable due to the tragic events. Our thoughts and prayers are with the affected family. Thank you for your understanding."
Statement on Kiwayu Safari Village's website
The Foreign Office website also warns: "If you visit Lamu Island, do so by air if possible. This is for security reasons and also because of the bad road conditions.
"Buses and other vehicles on the road to Lamu have been attacked by armed robbers in the past and overland travel from Lamu to Malindi should only be undertaken in an armed police convoy."
The Kiwayu resort's website had stated it takes "security and safety very seriously".
The website is now unavailable, posting the statement: "Sorry our website is unavailable due to the tragic events.
"Our thoughts and prayers are with the affected family. Thank you for your understanding.
Sent from my BlackBerry® smartphone provided by Airtel Nigeria.
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