Gateshead-born Harrison has been linked with Hartlepool United, who parted company with Dave Jones after dropping into the League Two relegation zone.
Harrison says he is flattered but insists his focus is on Sunday's Welsh Cup final against Bala Town.
"I've said from day one that I'm a very ambitious person," Harrison said.
"I want to manage at the very best level I can.
"It's nice to be linked to these jobs. But at this point in time the Welsh Cup Final is the be all and end all for me.
"Once that's finished we'll be preparing for the Champions League and then whatever will be will be."
Saints have already secured the Welsh Premier League title and the Nathaniel MG League Cup, Saints and earlier this season broke Ajax's 44-year world record for the most consecutive wins with 27 straight victories.
Harrison's side will achieve a third successive Welsh treble if they beat Bala in Sunday's Welsh Cup final in Bangor.

Thousands have been protesting against the takeover for a second day, in cities across Yemen.
On Friday, the Shia rebels said they were dissolving parliament and setting up an interim government.
The UN Security Council has warned of unspecified further steps if the group do not immediately return to talks.
The Houthis took control of parts of the capital Sanaa in September last year, forcing the resignation of President Abed Rabbo Mansour Hadi in January.
Defending the move, Houthi leader Abdul Malik al-Houthi said on Saturday that the declaration was "a historic, necessary and important step" to prevent a power vacuum in Yemen.
He added: "Our hand is extended to every political force in this country. The space is open for partnership, co-operation and brotherhood."
During protests in Sanaa on Saturday, Houthi gunmen fired into the air to disperse demonstrators. A bomb exploded outside the presidential palace, wounding three people.
There were also protests in at least three other Yemeni cities, with demonstrators calling the Houthis actions a coup.
Meanwhile, a rally in support of the Houthis took place in at a stadium in Sanaa.
The Gulf Cooperation Council, made up of Yemen's neighbours and led by Saudi Arabia, has also expressed alarm.
Iran has been accused of backing the Houthis, something both have denied, and analysts say the GCC's opposition signals Sunni Muslim hostility to the Houthis.
The Houthi's political takeover comes after a deadline they set for political parties to resolve the crisis expired, and UN-brokered peace talks failed.
Yemen has been riven by instability since protesters inspired by the Arab Spring forced the overthrow of President Ali Abdullah Saleh in 2011, who is believed to have been backing the Houthis.
The country is also fighting an al-Qaeda insurgency with the help of US drones. Despite the takeover, the US said it was continuing to work with Yemeni on counter-terrorism.
New Saints boss Craig Harrison has said he has ambitions to manage at "the highest level."
[NEXT_CONCEPT]
Yemen's Houthi rebels say they are open to working with other political factions, after taking over power in a move denounced by rivals as a coup.