That is quite easy to answer, not very credible at all, in fact, I have not found a single credible journalist or known source that has even mentioned the rumour.
For starters, Coutinho is reported to be on wages of 24m euros per year, for all the will in the world Arsenal simply will not pay that unless the player comes on a free transfer.
Secondly, Barcelona will want a significant fee for a player that they bought for over £140 Million and even though Arsenal has smashed the transfer budget myth I think we can all agree they do not have that sort of money to spend.
So from a basic financial perspective, there is no way that Arsenal can afford to sign Philippe Coutinho.
There is, of course, the loan option but think about that for a minute as well.
Barcelona would have to subsidise his wages and allow the Brazilian to leave on a free loan, does anyone seriously think that will happen?
Especially when you consider how much Barca have already spent this summer, they need to recoup some money, they are not immune to FFP.
But let’s look at a simpler loan scenario, let’s say Barca want £20 Million for a one year loan or £40 Million for two years and then Arsenal do pay his full wage, so let’s say for a two-year loan deal in fees and wages the total reaches £80 Million, why would they spend that on a player with no resale value because he is a loanee?
Break those figures down even further, say £10 Million per year loan fee and £15 Million in wages, for two years that is £50 Million for a player that cannot be sold.
For me that smacks of the old Arsenal, I just do not see this Arsenal doing that sort of deal, do you?
So, from a simple common-sense point of view, it is not credible in my opinion that Coutinho will be joining Arsenal in any form.