I believe that Kerensky told the truth on this point.
1. It fits with what we know from other sources.
Lord Stamfordham (George V's secretary) wrote on the 9th of March 1917: "I saw the Prime Minister (Lloyd George) this morning. He had not seen Sir George Buchanan's (British ambassador in Russia) telegram received last evening reporting his conversation with Mr Miliukov (new Foreign Minister), in which the latter urged the earliest possible departure of the Emperor from Russia, and suggested that the King and the British government should offer His Imperial Majesty an asylum in this country (...) No doubt the present Russian Government are anxious as to the safety of the Emperor and the Empress, and for that reason wish to expedite their departure."
King George V, Diary - 11 March : "Michael (Grand Duke, Miche-Miche) came to see me and we discussed the idea of poor Nicky coming to England."
Lord Stamfordham to A.J. Balfour (Secretary of State for Foreign Affairs) - 17 March: "The King has been thinking much about the Government's proposal that the Emperor Nicholas and his Family should come to England. As you are doubtless aware the King has a strong personal friendship for the Emperor, and therefore would be glad to do anything to help him in this crisis. But his Majesty cannot help doubting, not only on account of the dangers of the voyage, but on general grounds of expediency, whether it is advisable that the Imperial Family should take up their residence in this country."
A. J. Balfour to Lord Stamfordham - 20 March- Foreign Office, London: "His Majesty's Ministers quite realize the difficulties to which you refer in your letter, but they do not think, unless the position changes, that it is now possible to withdraw the invitation which has been sent, and they therefore trust that the King will consent to adhere to the original invitation, which was sent on the advice of His Majesty's Ministers."
Lord Stamfordham to A.J. Balfour - 24 March - Windsor Castle: "Every day the King is becoming more concerned about the question of the Emperor and the Empress of Russia coming to this country. His Majesty receives letters from people in all classes of life, saying how much the matter is being discussed, not only in Clubs but by working men, and that Labour Members in the House of Commons are expressing adverse opinions to the proposal (...) The King desires me to ask you whether after consulting the Prime Minister, Sir George Buchanan should not be communicated with a view to approaching the Russian Government to make some other plan for the future residence of Their Imperial Majesties?"
Same day - "The King wishes me to write again on the subject of my letter of this morning. He must beg you to represent to the Prime Minister that from all he hears and reads in the Press, the residence in this country of the Ex-Emperor and Empress would be strongly resented by the public, and would undoubtedly compromise the position of the King and Queen from whom it is already generally supposed the invitation has emanated.
Buchanan ought tio be instructed to tell Milyukov that the opposition to the Emperor and the Empress coming here is so strong that we must be allowed to withdraw from the consent previously given to the Russian Government's proposal."
Lord Stamfordham, note of meeting - 28 March - Windsor Castle: "I saw the Prime Minister (Lloyd George) at 10 Downing Street, and tried to impress upon him the King's strong opinion that the Emperor and Empress of Russia should not come to this country, and that the Government ought to inform Monsieur Miliukov that since they had agreed to his proposal that Their Imperial Majesties should take up their residence in this country public opinion here had become so stoutly opposed to the idea that His Majesty's Government must withdraw the consent previously given.
(...) I afterwards saw Mr Balfour and called attention to a telegram which I had just seen from Sir George Buchanan, who in his conversation with Monsieur Miliukov evidently took it for granted that the Emperor and Empress were coming to England, and that it was only question of delay with regard to certain matters that had not been cleared up, which prevented an early start.
I told Mr Balfour that after what the King had written to him His Majesty expected that Sir George Buchanan would by now have been informed that the whole question was being reconsidered, and that our previous Agreement could no longer be held as binding. Mr Balfour said that he would draft a telegram to Sir George Buchanan this afternoon and send it to the Prime Minister for his approval."