The Saints beat Decic 4-1 on aggregate in the opening round of Champions League qualifying this summer before losing 7-1 over two legs to reigning Hungarian champions Ferencvaros in round two of qualifying.
Harrison’s side were then beaten 1-0 by Petrocub on aggregate in the Europa League third qualifying round, but cruised to a 3-0 victory away over FK Panevėžys before drawing the second leg 0-0 to book their place in the Europa League Conference group stage.
Former Wales midfielder Owain Tudur Jones told Radio Wales Breakfast that it would be “no big drama” if TNS played their European games at Shrewsbury.
“TNS play their home games in England anyway in Oswestry, so there won’t be that much of a change regardless if it is Shrewsbury,” he said.
“They have to do what’s right for the club, there’s going to be people saying ‘why not play the games in Wrexham?’, I don’t think Wrexham is a possibility from what we hear, that (Shrewsbury) would be the number one (option) probably.
“Cardiff is far away, European football has been played at Cardiff City Stadium over the years by various clubs, TNS being one of them, and it is three hours away from what would be classed as their ‘home’, so Shrewsbury for me is not the end of the world, no big dramas.
“Perhaps the FAW (Football Association of Wales) would disagree with that, but as a club they’ll be looking at all avenues, best -laid plans for them and if that’s Shrewsbury then so be it.”
Having seen his side make history, Harris hopes the Saints’ success can help raise the profile of domestic football in Wales.
“I have to say, I’ve always been competitive, and I like looking at things differently to other people,” Harris added.
“I saw an opportunity in Welsh football, and I backed it.
“I like to think now that the Welsh public will start getting behind us and their local Welsh teams. Hopefully, we can all try and push the league forward.”
A plum tie with Italian giants Fiorentina is arguably the greatest reward for Harris’ investment into the Saints’ during his tenure as chairman of the club.
On the trip to Italy, Harris said: “Now that’s going to be some fun.
“I like that area of the world, I’ve travelled around there a few times, it’s a lovely place.
“They have some great wine and great food. I’ll come back fatter than when I went.”
Dates and kick-off times for the Saints’ six group stage matches will be confirmed by Uefa on Saturday, 31 August.