Trinidadian Simmons, who has led the Irish since 2007, is believed to have been interviewed for the job prior to the current World Cup.
BBC Sport has been told that Simmons, 51, has been offered the West Indies post but that negotiations have yet to be successfully concluded.
Simmons's Irish exited from the World Cup after Sunday's defeat by Pakistan.
The coach is under contract with Ireland until December but Cricket Ireland are not expected to stand in his way if he decides to accept the West Indies offer.
A quick Simmons appointment would see him take charge for the Test series against England in the Caribbean next month.
Stuart Williams has been in temporary charge of the Windies since they parted company with Ottis Gibson last August.
Simmons, who played 26 Tests from 1988 until 1997, has been linked with the West Indies job on a number of previous occasions.
And his stock has risen further over the last month given Ireland's impressive performances at the World Cup.
Since replacing Adi Birrell after the 2007 World Cup, Simmons has guided Ireland to the 2011 and 2015 tournaments and earned wins over Test nations England, the West Indies and Zimbabwe.
Simmons's Ireland squad defeated the West Indies in their opening World Cup game in Nelson last month but missed out on a quarter-final spot on run-rate despite further group victories over the United Arab Emirates and Zimbabwe.
During Simmons's tenure, he has guided Ireland to six successive ICC global tournaments while his side have also won 10 trophies contested by Associate countries.

Even though Cuba is less than 100 miles away from the US, the two counties fell out in 1959, and no American president has visited since then.
The fall out was because two brothers called Fidel and Raul Castro took power in a revolution.
America did not agree with how they ran the country.
Since being president, Obama has tried to make America's relationship with Cuba more friendly, and in 2015 he re-opened the US embassy which was closed 54 years earlier.

Patrick Bauer and Nathan Byrne helped Karl Robinson's men complete a perfect smash and grab, cancelling out Zach Clough's free-kick.
A victory which furthers the Addicks' own play-off ambitions was made all the more impressive considering that they were a man short from the 12th minute onwards following Lewis Page's red card.
After Page was sent off for a professional foul on Josh Vela, Clough then made the punishment count double with a brilliant free-kick, his ninth goal of the season.
But Robinson's side refused to lie down, equalising on 23 minutes when Jake Forster-Caskey's free-kick was turned in at the far post by defender Bauer.
The Addicks then grabbed a shock lead in first-half stoppage time when Forster-Caskey led an impressive counter, feeding Byrne to slot the ball under the advancing Ben Alnwick.
Bolton, now without a win in their last five games in all competitions, threw all they had at Charlton after that and struck the woodwork when Andrew Taylor's cross deflected off Chris Solly.
They failed to seriously test goalkeeper Declan Rudd in the second half, however, as they slipped to a second consecutive 2-1 defeat in front of their own fans.
Charlton, unbeaten since Christmas, have now picked up 11 points from their last five matches to climb to 11th, just six points off a play-off place.
Report supplied by the Press Association.
Ireland coach Phil Simmons is mulling over whether to accept an offer to take charge of his native West Indies.
[NEXT_CONCEPT]
President Obama has arrived in Cuba, an island in the Caribbean, for an historic three day visit.
[NEXT_CONCEPT]
Charlton Athletic came from behind with 10 men to deliver a blow to Bolton Wanderers' League One promotion hopes  at the Macron Stadium.