Jorah clearly feels great affection and loyalty for Daenerys. Are his feelings romantic or sexual in nature? Or does he see himself as a friend, counsellor, or father figure?
The age gap between them could be a barrier to romantic feelings on Jorah's part. Jorah is aged about 50, whereas Daenerys is in her early 20s in the television show and only 13 in the books.
Yes! and he's even blatant about it too (in the books).
On many occasions Jorah shows signs of not only loving Daenerys as his sovereign and her being the Mother of Dragons, he also shows signs of having feelings towards her, romantically.
He confesses to Dany that his second wife Lynesse Hightower looks like Dany, which Dany realises that it explains a lot:
She gave him leave to go, but as he was lifting the flap of her tent, she could not stop herself calling after him with one last question. “What did she look like, your Lady Lynesse?”
Ser Jorah smiled sadly. “Why, she looked a bit like you, Daenerys.” He bowed low. “Sleep well, my queen.”
Dany shivered, and pulled the lionskin tight about her. She looked like me. It explained much that she had not truly understood. He wants me, she realized. He loves me as he loved her, not as a knight loves his queen but as a man loves a woman.
-A Song of Ice and Fire: Book Two - A Clash of Kings, Chapter Twelve (Daenerys I).
In A Storm of Swords he grabs her and kisses her when they're alone and tells her that he loves her and that they should get married, but she kindly walks him straight back to his Friendzone:
“What if Captain Groleo refuses to change course, though? And Arstan, Strong Belwas, what will they do?”
Ser Jorah stood. “Perhaps it’s time you found that out.”
“Yes,” she decided. “I’ll do it!” Dany threw back the coverlets and hopped from the bunk. “I’ll see the captain at once, command him to set course for Astapor.” She bent over her chest, threw open the lid, and seized the first garment to hand, a pair of loose sandsilk trousers. “Hand me my medallion belt,” she commanded Jorah as she pulled the sandsilk up over her hips. “And my vest -” she started to say, turning.
Ser Jorah slid his arms around her.
“Oh,” was all Dany had time to say as he pulled her close and pressed his lips down on hers. He smelled of sweat and salt and leather, and the iron studs on his jerkin dug into her naked breasts as he crushed her hard against him. One hand held her by the shoulder while the other slid down her spine to the small of her back, and her mouth opened for his tongue, though she never told it to. His beard is scratchy, she thought, but his mouth is sweet. The Dothraki wore no beards, only long mustaches, and only Khal Drogo had ever kissed her before. He should not be doing this. I am his queen, not his woman.
It was a long kiss, though how long Dany could not have said. When it ended, Ser Jorah let go of her, and she took a quick step backward. “You... you should not have...”
“I should not have waited so Iong,” he finished for her. “I should have kissed you in Qarth, in Vaes Tolorro. I should have kissed you in the red waste, every night and every day. You were made to be kissed, often and well.” His eyes were on her breasts.
Dany covered them with her hands, before her nipples could betray her. “I... that was not fitting. I am your queen.”
“My queen,” he said, “and the bravest, sweetest, and most beautiful woman I have ever seen. Daenerys -
“Your Grace!”
“Your Grace,” he conceded, “the dragon has three heads, remember? You have wondered at that, ever since you heard it from the warlocks in the House of Dust. Well, here’s your meaning: Balerion, Meraxes, and Vhagar, ridden by Aegon, Rhaenys, and Visenya. The three-headed dragon of House Targaryen - three dragons, and three riders.”
“Yes,” said Dany, “but my brothers are dead.”
“Rhaenys and Visenya were Aegon’s wives as well as his sisters. You have no brothers, but you can take husbands. And I tell you truly, Daenerys, there is no man in all the world who will ever be half so true to you as me.”
-A Song of Ice and Fire: Book Three - A Storm of Swords, Chapter Eight (Daenerys I).
He tells her so again when she banishes him:
“I protected you. I fought for you. Killed for you.”
Kissed me, she thought, betrayed me.
“I went down into the sewers like a rat. For you.”
It might have been kinder if you’d died there. Dany said nothing. There was nothing to say.
“Daenerys,” he said, “I have loved you.”
And there it was. Three treasons will you know once for blood and once for gold and once for love.
-A Song of Ice and Fire: Book Three - A Storm of Swords, Chapter Fourty-Two (Daenerys VI).
But to no avail, she banishes him and he embarks on his quest to escape the dreaded Friendzone; and fails miserably, even finds a whore in the Volantene whorehouse who looks like Dany. Oh, poor Jorah!
But this is why he goes through the trouble of trying to take Tyrion back to Dany, he thinks that it will win back her trust for him.
