Yes... but it sort of depends on how you define the term 'lover'. Some NSFW poetry follows below.

There are three men who are usually put forward as candidates for James' lovers, the third of whom is most supported by evidence.

The first man who is usually termed James' first romantic male interest was Esmé Stuart, later the Earl of Lennox. The exact nature of their relationship has never been confirmed, but even as it unfolded, James’ contemporaries were concerned about its nature. One clergyman is on the record as remarking ‘the Duke of Lennox went about to draw the King to carnal lust’, and it was often remarked that the two were openly physically affectionate. However, importantly, James was only 14 when their relationship - however it was defined - began, and so the exact nature of their interactions has been theorised as being anything from paternal to borderline paedophilic. Stuart converted from Catholicism to please James, and James made him first the gentleman of the bedchamber and then finally the Earl of Lennox. This pattern of bestowing titles upon his ‘favourites’ became a real point of contention at James’ court. Lennox was later exiled at the request of several Scottish lords, and died in 1583.

His next affair was with a young man named Robert Carr, who impressed the king in 1607 by falling off a horse and breaking his leg. No, really. A courtier later wrote that Carr was granted ‘all favours’, with the king ‘teaching him Latin every morning’, which might be euphemistic or it might not be - we just don't know. Carr, like Stuart before him, was made a gentleman of the bedchamber, writing in a letter that Carr ‘deserved more trust and confidence of me than ever man did’. The relationship ended fairly spectacularly when Carr was accused of blackmail.

James’ most well-known affair was with George Villiers, who was a relatively unimportant man in terms of social standing. They met in 1614 and by 1615, Villiers had been made a knight, followed by the title of Duke of Buckingham in 1623. We know the most about James’ relationship with Villiers because several contemporary sources survive, including letters between the two men and satirical poems and statements written about them by their political detractors. For example, the French poet Théophile de Viau wrote a poem in protest at Villiers’ dukedom (my own very literal translation, so apologies in advance for the lewdness):

Apollo with his songs

Debauched the young Hyacinthus,

If Corridon fucks Amyntas,

Caesar loved only boys.

One man fucks the Baron of Bellegarde

Another fucks the Count / Earl of Tonnerre.

And this learned King of England,

Did he not fuck the Duke of Buckingham?

I have neither the status nor the rank

Which makes a Marquis of a wench.

And yet, you know I fuck

As well as any Prince of royal blood.

We have many extant letters written between James and Villiers, in which they address each other as ‘husband’ and ‘wife’, as well as ‘sweet child’ and ‘dad’, which of course does not carry all of the same homoerotic undertones as certain sexual subcultures today, but is nonetheless slightly suggestive. For example, here is an example of some of the text from a letter written by James:

I desire only to live in this world for your sake… I had rather live banished in any part of the Earth with you than live a sorrowful widow’s life without you… God bless you, my sweet child and wife, and grant that ye may ever be a comfort to your dear dad and husband.

And another example:

I naturally so love your person, and adore all your other parts, which are more than ever one man had, that were not only all your people but all the world besides set together on one side and you alone on the other, I should to obey and please you displease, nay, despise them all.

And a letter from Villiers to James:

sir, all the way hither I entertained myself, your unworthy servant, with this dispute, whether you loved me now… better than at the time which I shall never forget at Farnham, where the bed’s head could not be found between the master and his dog

Villiers was there when James died in 1625.

So, the evidence that suggests they were lovers:

• as mentioned, there were substantial rumours surrounding James and his male favourites during his lifetime. Much court gossip focused on the potentially salacious nature of James' interactions with men. Rumour, of course, is not evidence that he was romantically involved with men, but it is at least suggestive that there was enough evidence at the time to stir the pot of speculation. One courtier is quoted as saying '‘in wanton looks and wanton gestures they exceeded any part of womankind. The kissing them after so lascivious a mode in public and upon the theatre, as it were, of the world prompted many to imagine some things done in the tyring house that exceed my expression no less than they do my experience'.
• the letters that James and Villiers wrote, some quoted above, which have been collected and are published in a book (Bergeron, David M (1999). King James and Letters of Homoerotic Desire. Iowa: University of Iowa Press)
• there is a famous passageway at Apethorpe Hall, a property used by James, which is widely believed to have linked the bedroom he used to the bedroom of George Villiers. This is, for fairly obvious reasons, often taken as evidence of sexual, or at least romantic, liaisons.

And there are, of course, a few problems with all the evidence:

• as mentioned, rumour itself is not evidence.
• James was vocally anti sodomy and in fact included it as one of the ‘horrible crimes which ye are bound in conscience never to forgive’ as a king, singling it out as a crime that judges were never to pardon in any circumstance. Some people who do accept that he had affairs with men have here found evidence of hypocrisy, but it may be more complicated than that. The fact is that we don’t know that James would have been guilty, as he would have termed it, of sodomy himself. Although vocally opposed to sodomy, he did not oppose any less physical romantic relationship between men, or indeed a physical relationship which did not include sodomy, so in theory, it could have been a case of anything else goes, or it could be a case of James being so vocally opposed to same sex sexual activity that he was able to engage in deep platonic relationships with men without fear of traversing the boundary between the two. It really is speculation either way.
• James was notoriously pious. Even now, he's best known for the King James Bible. This, of course, puts something of a spanner in the works; the Bible does contain passages which have been received as being anti homosexuality. It isn't a stretch to imagine that James might have shared this viewpoint. However, there is another possibility, which is that James read the Biblical text as being anti sodomy rather than anti homosexual acts; this reading is still one which is proffered as an alternative to the more common 'the Bible speaks out against homosexuality' message. There is some evidence that could be used to illustrate this, which is James' speech about Villiers:

I, James, am neither a god nor an angel, but a man like any other. Therefore I act like a man and confess to loving those dear to me more than other men. You may be sure that I love the Earl of Buckingham more than anyone else, and more than you who are here assembled. I wish to speak in my own behalf and not to have it thought to be a defect, for Jesus Christ did the same, and therefore I cannot be blamed. Christ had John, and I have George.

• there are explanations for the tunnel at Apethorpe, ranging from the fact that we do not have conclusive proof of whose bedrooms they were to the idea that, as his companion, it would be expected for Villiers to have access to James' bedroom. The counter argument to that would be that, as it was expected, there would be no need for a secret tunnel - but either way, that can't be discounted as a real issue with the evidence.

So, in short, yes, there is plenty of evidence to suggest that James VI of Scotland and I of England had male lovers, but the degree to which any of these relationships were physical is unknown, and much of the evidence is either speculation, rumour, or open to interpretation.

Sources:

• Bergeron, David M. (1999), King James and Letters of Homoerotic Desire, Iowa City: University of Iowa Press
• Crompton, Louis (2006). Homosexuality and Civilization. USA: Harvard University Press
• Stewart, Alan (2003). The Cradle King: A Life of James VI & I. London: Macmillan