The Gulf island initially staged the curtain-raiser in 2006, standing in for Australia and Melbourne as the city did not want the race to clash with their hosting of that year's Commonwealth Games.
Bahrain then pushed for the opening slot, but they had to pay for the privilege as F1 supremo Bernie Ecclestone upped their race fee from 25million US dollars (£16.2million) to 40million US dollars (£26million).
It was a price Bahrain was willing to pay, and they were granted the honour of ushering in the season from 2010.
In 2011, however, anti-government demonstrations that resulted in the deaths of a number of protesters led to the postponement of the race, and eventually its withdrawal from the calendar that year.
Bahrain returned to F1 last season, albeit demoted to the fourth race on the calendar, as it is this year. With another three years remaining on their current contract with Ecclestone, up to and including 2016, Alzayani would love for his circuit to take over the season-opener from Melbourne.
Asked for his preferred position on the calendar, Alzayani replied: "Personally, first race.
"It's a good start to the season, it gives us the chance to have the teams here longer, there is more anticipation, more unknowns.
"You don't know how the cars will react to the new tyres, what the new regulations will do and the drivers are just back into their rhythm.
"But we are ready to have the race, whether we're first, second, third or fourth on the calendar, and there are pros and cons for each.
"The decision is not entirely ours. We are talking about 2014 onwards, but it's a bit early to talk about the calendar for 2014.
"We're open to suggestion, and whether it's the first race or fourth for us, we're ready.
"We'll come to that towards the end of the season when the calendar is being discussed and decided.
"There are some people who want the first race, and there are some people who don't want the first race. We'll see how it goes.
"It's really a discussion between ourselves and Bernie."
In 2011 Bahrain was also due to hold the final pre-season test ahead of the opening grand prix. With complaints from the teams this pre-season about the cold and wet weather in Jerez and Barcelona being far from ideal, Alzayani feels Bahrain could again help out in the future.
"It's on the table," added Alzayani.
"The only problem we have - not us, but the teams - is the cost of the logistics in coming here and going back to Europe.
"But that is one of the advantages of having the first race, in that they can come to the last test and then leave most of their equipment here until race weekend.
"This was the plan in 2011. We were supposed to have the test at the end of February, and then the race on March 13.
"So if we go after the first race, then we will go after a test as well, but again that is not our decision. It is the teams who decide where to test."
Supremo Bernie Ecclestone is not opposed to the idea of Bahrain again becoming the opening race, as he said: "We could do I suppose.
"We changed because Australia has been around a long time and they wanted to maintain that.
"I hear they (Bahrain) are quite keen. We need to have a good look at it."