Gregg Ryder has been linked as a potential replacement for Ray McKinnon as boss of the Bairns, the Falkirk Herald understands.
The 31-year-old has top flight managerial experience in Iceland and worked alongside Hallgrimsson at club level before his ascent to the national team job where he led Iceland to a memorable Euro 2016 win over England.
Following their alliance Ryder, the youngest boss to coach in the country, went on to manage IBV, where he gained Europa League experience and also steered Throttur Reykjavik into the top flight during for four and a half years at the helm.
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The Newcastle-born boss was close to landing the top job at Gateshead before former Newcastle midfielder Steve Watson was appointed and was also on the shortlist at Morton to fill their vacancy, a position eventually assumed by future Falkirk boss McKinnon.
Ryder is on record as keen to move to Scotland and told his local paper in the north-east of England "I've spent a long time away from home and I now need to a club in Britain to give me an opportunity. A club is Scotland could be a good option."
The Bairns however are in no rush to appoint a manager and revealed last week more than 30 applications had been received. It is also known former Kilmarnock and Forth Valley Academy boss Kenny Shiels is interested in the post.
Lee Miller and David McCracken currently hold the position of interim managers and made a strong start when they led the side to a 4-1 win over Linlithgow Rose in the William Hill Scottish Cup and followed it up with a 3-0 home win over Stranraer on Saturday.
Other names linked with the role by bookmakers in the wake of McKinnon's dismissal included East Fife's Darren Young, Rangers assistant Gary McAllister, former boss John Hughes and Edinburgh City's James McDonaugh.